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Research ArticleRegional Frequency Analysis of Extremes Precipitation UsingL-Moments and Partial L-Moments
Said Arab Khan12 Ijaz Hussain1 Tajammal Hussain3 Muhammad Faisal45
Yousaf ShadMuhammad1 and Alaa Mohamd Shoukry67
1Department of Statistics Quaid-i-Azam University Islamabad Pakistan2Government Degree College Lahore Swabi Khyber Pakhtunkhwa Pakistan3Department of Statistics COMSATS Institute of Information Technology Lahore Pakistan4Faculty of Health Studies University of Bradford Bradford UK5Bradford Institute for Health Research Bradford Teaching Hospitals NHS Foundation Trust Bradford UK6Arriyadh Community College King Saud University Riyadh Saudi Arabia7Workers University Cairo Egypt
Correspondence should be addressed to Ijaz Hussain ijazqauedupk
Received 21 December 2016 Revised 26 January 2017 Accepted 7 February 2017 Published 7 March 2017
Academic Editor Mouleong Tan
Copyright copy 2017 Said Arab Khan et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Extremes precipitation may cause a series of social environmental and ecological problems Estimation of frequency of extremeprecipitations and its magnitude is vital for making decisions about hydraulic structures such as dams spillways and dikes In thisstudy we focus on regional frequency analysis of extreme precipitation based on monthly precipitation records (1999ndash2012) at 17stations of Northern areas and Khyber Pakhtunkhwa Pakistan We develop regional frequency methods based on L-moment andpartial L-moments (L- and PL-moments) The L- and PL-moments are derived for generalized extreme value (GEV) generalizedlogistic (GLO) generalized normal (GNO) and generalized Pareto (GPA) distributions The 119885-statistics and L- and PL-momentsratio diagrams of GNO GEV and GPA distributions were identified to represent the statistical properties of extreme precipitationin Northern areas and Khyber Pakhtunkhwa Pakistan We also perform a Monte Carlo simulation study to examine the samplingproperties of L- and PL-moments The results show that PL-moments perform better than L-moments for estimating large returnperiod events
1 Introduction
Hydraulic and hydrologic designs are key steps in planningof any water project Any problem pitched at designingstage will result in the failure of design irrespective of thefact how correctly the other steps are taken Hydrologistsdeal with water-related issues problems of quantity qualityand availability in the society that known as hydrologicevents Stochastic methods are often used to understandsources of uncertainties in physical processes that give riseto observed hydrologic events as precipitation and streamflow estimates depend on the past or future events Severalstatistical methods offered to minimize and summarize theuncertainties of observed data and frequency analysis is oneof them It determines that how often a particular event will
occur by estimating the quantile 119876119879 for return period of 119879where 119876 is the magnitude of the event that occurs at a giventime and location
Dalrymple [1] proposed regional frequency analysis(RFA) method for pooling various data samples It is index-flood procedure in hydrology Hosking et al [2] studiedthe properties of probability-weighted moments (PWMs)method based on RFA method This method is first used byGreis and Wood [3] and Wallis [4] Cunnane [5] reviewedtwelve methods of RFA and related regional PWMs algo-rithm
Initially PWMs are considered as an alternative parame-ter estimation method however it was difficult to interpretdirectly as measures of the shape and scale parameters ofdistribution RFA can forecast the flood flow using empirical
HindawiAdvances in MeteorologyVolume 2017 Article ID 6954902 20 pageshttpsdoiorg10115520176954902
2 Advances in Meteorology
formula and unit-hydrograph procedure Subramanya [6]and it can also estimate the quantiles of extreme precipitation
Hosking and Wallis [7] showed that RFA method basedon L-moments is used to detect homogeneous regions toselect suitable regional frequency distribution and to predictextreme precipitation quantiles at region of interest
Whilst L-moment methodology is effective in estimatingparameters it may not valid for predicting high returnperiod events Wang [8] suggested that relatively small floodsmight create disturbance in the analysis To overcome suchsituation a censored sample can be used as by Cunnane [9]
Wang [8] proposed partial probability-weightedmoments (PPWMs) for fitting the probability distributionfunction to the censored sample Partial L-moments(PL-moments) are variants of the L-moments and similar toPPWMs PL-momentsmethod has used in fitting generalizedextreme value (GEV) distribution for censored flood samples(see Wang [8 10 11] and Bhattarai [12]) Bhattarai [12] foundthat censoring flood samples are nearly thirty percent ofbasic L-moments
Shabri et al [13] used Trim L-moments (TL-moments)for the RFA and compared its performance with L-momentsSaf [14] determined hydrologically homogeneous region andregional flood frequency estimates by using index-floodtechnique along with L-moments for the West Mediter-ranean River Turkey L-moments method is also used toassign a suitable regional distribution for the individualsubregions and to assess their homogeneity see Abolverdiand Khalili [15] Hussain and Pasha [16] suggested theregional flood frequency analysis based on L-momentsTheyused discordancy measure for data screening and used thefour-parameter Kappa distribution with 500 simulationsfor the heterogeneity analysis Zakaria et al [17] used thePL-moments technique and found another link for thehomogeneity analysis Shahzadi et al [18] showed that thegeneralized normal (GNO) distribution is suitable for theregional quantile estimation at maximum return period andthe GEV distribution for the overall regions at low returnperiod based on relative RMSE and relative absolute biasMost commonly used statistical distributions for high climatemodeling are as follows the logistic distribution with threeparameters lognormal distribution with three parametersLog Pearson type III GEV and generalized Pareto (GPA)(Coles [19] Katz et al [20] Abida and Ellouze [21] Feng et al[22] Yang et al [23] Villarini et al [24] Zakaria et al [17] Sheet al [25])Moreover GEV andGPAdistributions are suitableif data contains extremes values Over the years the GNOGEV generalized logistic (GLO) and GPA distributionshave been widely employed in the extreme value estimationof annual flood peaks In this study we aim to developRFA method based on L- and PL-moments approach Ourproposed method can be used at all levels of regional analysissuch as identification of homogeneous regions identificationand also testing of the suitable probability regional frequencydistribution based on 119885-statistic and the L- and PL-momentratio diagram and estimation of the flood quantile at siteof interest We use 17 sites of Northern areas and KhyberPakhtunkhwa as a case study to perform the analysis
We explain the methodology of the L-moments and PL-moments in Section 22 Section 23 shows the applicationof the RFA where we choose the appropriate distributionfor the regional analysis Section 33 provides the estimationof quantile for both the small and large return period Theresults of our simulations studywill be presented in Section 4
2 Material and Methods
21 Study Area and Data Sources The data was collectedfor this study from Karachi Data Processing Center throughPakistan Meteorological (PMD) Islamabad Monthly precip-itation data has been recorded from 1999 to 2012 There are17 meteorological stations of Northern areas and KhyberPakhtunkhwa Pakistan These stations are full precipitationregions that affect areas in Pakistan where water is essentialfor hydropower and flood plains Figure 1 shows the locationof the study area and geographic distribution of precipitationstations There is no missing value in this data set
22 Statistical Methods
221 The L-Moments Conventional moments method maybe used for estimating the parameters of probability distri-bution However this approach has some serious drawbacksRatio ofmoment estimators is biased and often assumption ofbeing normally distributed is violated Wallis et al [26] Fur-thermore it is sensitive to outliers Pearson [27]Therefore itis unreliable for skewed distributions
Hosking [28] proposed L-moments approach to over-come the above problems The L-moments may preciselydescribe the statistical properties of hydrological informa-tion and it can write as a linear function of the PWMs ThePWMs of order 119903 were properly described by Greenwood etal [29] as
where 119865 = 119865(119909) is the cumulative distribution function of arandom variable 119909 and 119903 is a nonnegative integer of the realnumber that is 119903 = 0 1 2 3 Therefore the first four L-moments which are the linear combinations of the PWMsare
1205821 = 12057301205822 = 21205731 minus 12057301205823 = 1205732 minus 61205731 + 12057301205824 = 201205733 minus 301205732 + 121205731 minus 1205730
Figure 1 Locations of Northern areas and Khyber Pakhtunkhwa and the meteorological stations (119909 axis indicates Longitude [E] 119910 axisindicates Latitude [N])
The L-moments have no units of measurement which arecalled the L-moments ratios The L-moments ratios pro-posed by Hosking [28] are computed as
120591 = 12058221205821
1205913 = 12058231205822
1205914 = 12058241205822 (5)
4 Advances in Meteorology
where 120591 represents the L-coefficient of variation (L-Cv)1205913 represents the L-coefficient of skewness (L-Cs) and 1205914represents the L-coefficient of kurtosis (L-Ck)
The arranged sample is given as 119909(1) le 119909(2) le 119909(3) sdot sdot sdot le119909(119899) Wang [8] stated that the statistic
119887119903 = 1119899119899sum119894=1
(119894 minus 1) (119894 minus 2) sdot sdot sdot (119894 minus 119903)(119899 minus 1) (119899 minus 2) sdot sdot sdot (119899 minus 119903)119909(119894) (6)
is an unbiased estimator of 120573119903So
1198971 = 11988701198972 = 21198871 minus 11988701198973 = 61198872 minus 61198871 + 11988701198974 = 201198873 minus 301198872 + 121198871 minus 1198870
(7)
where 1198971 1198972 1198973 and 1198974 are the first four L-moments of thesample And similarly
222 The Partial L-Moments Wang [8 10 11] introduced aconcept of partial probability-weighted moments (PPWMs)that will estimate the higher quantiles of flood flows Data canbe censored to the right tail or left tail
Initially PPWMswere to take out the smaller values fromthe process of distribution fitting because such values haveslight influence on the frequency analysis and are nuisance tothe fitting process
The left tail PPWMs are defined by Wang [8 11] as
120573lowast119903 = int11198650
119909 (119865) 119865119903119889119865 (9)
where 1198650 = 119865(1199090) which is the lower limit of the censoredobservations and 1199090 is the censoring threshold value
The PPWMs elongated form described by Wang [10] areto be given a censored sample as
120573119903 = 11 minus 119865119903+10 int11198650
119909 (119865) 119865119903119889119865 (10)
If the value of 1198650 is starting from the zero then the result ofPPWMswill be the same as the usual PWMs As 119909(1) le 119909(2) le119909(3) sdot sdot sdot le 119909(119899) is the arranged sample Wang [10] describes theunbiased estimator of 120573119903 as
119903 = 1(1 minus 119865119903+10 ) 119899119899sum119894=1
(119894 minus 1) (119894 minus 2) sdot sdot sdot (119894 minus 119903)(119899 minus 1) (119899 minus 2) sdot sdot sdot (119899 minus 119903)119909lowast(119894) (11)
where
119909lowast(119894) = 0 for 119909(119894) le 119909(0)119909lowast(119894) = 119909(119894) for 119909(119894) gt 119909(0) (12)
The censoring level 1198650 is the prior selection of the numberof censored sample data The procedure that determines thenumber of sample data points are to be censored
1198650 = 1198990119899 (13)
where 1198990 and 119899 are the lengths of sample which are to becensored and uncensored respectively Similarly 1199090 is thehighest value of the censored sample The first four PL-moments for the PPWMs are
1 = 12057302 = 2 1205731 minus 12057303 = 1205732 minus 6 1205731 + 12057304 = 20 1205733 minus 30 1205732 + 12 1205731 minus 1205730
(14)
Similarly the PL-moments ratios can be written as
120591 = 21
1205913 = 32
1205914 = 42
(15)
where 120591 1205913 and 1205914 denote the partial L-coefficient of variation(PL-Cv) partial L-coefficient of skewness (PL-Cs) and par-tial L-coefficient of kurtosis (PL-Ck) respectively Thereforethe first four sample PL-moments can be computed as
1198971 = 01198972 = 21 minus 01198973 = 62 minus 61 + 01198974 = 203 minus 302 + 121 minus 0
(16)
And the first four sample PL-moments ratios can be com-puted as
where 119905 1199053 and 1199054 represent the sample partial L-momentsratios of the PL-Cv PL-Cs and PL-Ck respectively Thederivation is L-moment and PL-moments are given inAppendix In the present study different level of censoringthreshold is selected
23 Regional Frequency Analysis Hosking and Wallis [7 30]identified the following four steps to explain the procedure ofthe RFA
(1) Data screening(2) Designing of the homogeneous region(3) Selection of an appropriate probability distribution(4) Parameters estimation of the appropriate probability
distribution
231 Data Screening We screened data anomalies beforeapplying any statistical analysis
232 Discordance Test Hosking and Wallis [7] suggested adiscordancy measure (119863119894) test that recognizes the locationswhere sample L-moments are marked contrarily from themost other locations Locations with the large flaws in thedata will be flagged as discordant
The discordancy test for a region containing119873 locationsfor site 119894 is proposed by Hosking and Wallis [7] as follows
119863119894 = 13119873 (119906119894 minus 119906)119879 119878minus1 (119906119894 minus 119906) 119894 = 1 2 119873 (18)
where 119906119894 is the vector containing the three sample L-momentsratios for the site 119894 expressed as
119906 is the average vector of 119906119894 for the overall region that is
119906 = 1119873119873sum119894=1
119906119894 (20)
and 119878 is the covariance matrix for the sample that can beexpressed as
119878 = 119873sum119894=1
(119906119894 minus 119906) (119906119894 minus 119906)119879 (21)
Broadly speaking a location or a site is considered to bediscordant from the whole region or group if the value of 119863119894is larger than the critical value
233 Heterogeneity Test The homogeneity measure (119867119895)identifies homogenous regions Hosking and Wallis [7] Itis also useful to tag the locations if they are plausible tohandle as a homogeneous region It estimates the amountof heterogeneity in the overall region The heterogeneity test(119867119895) is computed as
where 120583V119895 and 120590V119895 are representing the mean and standarddeviation of the simulated 119881119895 values Also
1198811 = 119873sum119894=1
119899119894 (119905(119894)2 minus 1199051198772 )sum119873119894=1 119899119894
12
1198812 =sum119873119894=1 [119899119894 (119905(119894)2 minus 1199051198772 )2 + (119905(119894)3 minus 1199051198773 )212]
sum119873119894=1 119899119894
1198813 =sum119873119894=1 [119899119894 (119905(119894)3 minus 1199051198773 )2 + (119905(119894)4 minus 1199051198774 )212]
sum119873119894=1 119899119894
(23)
Here 1199051198772 1199051198773 and 1199051198774 are region average L-moments or PL-moments ratios We assessed the heterogeneity of a region assuggested by Hosking and Wallis [7]
Region is acceptably homogeneous if119867 lt 1Region is possibly heterogeneous if 1 le 119867 lt 2Region is definitely heterogeneous if119867 ge 2
24 Selection of the Appropriate Probability DistributionHosking andWallis [7] proposed two approaches to select thedistribution that fitted best the data the L-moment ratios dia-gram and the 119885-test The L-moment ratios diagram is usingthe unbiased estimators Hosking [28] Stedinger et al [31]Vogel and Fennessey [32] and Hosking [33]The L-momentsratio diagram is a plot of the computed values L-Cs andthe observed values L-Ck of the distribution function Thecurves indicate the hypothetical connections between L-Csand L-Ck of the candidate distribution The L-moment ratiodiagrams have been proposed for discriminating between thecandidate probability distributions in describing the regionalinformation (Hosking [28] Stedinger et al [31] Hoskingand Wallis [7]) The L-moments ratio diagrams have beenused as a component of probability distribution process forregional information (Schaefer [34] Pearson [35] Vogel andFennessey [32] Vogel et al [36] Chow and Watt [37] OnOzand Bayazit [38] Vogel and Wilson [39] Peel et al [40])
Hosking and Wallis [7] suggested a measure to see howwell the L-Cs and L-Ck of the fitted probability distributionmatch the regional average L-Cs and L-Ck of the observedinformation
The measure goodness of fit for every single selectedprobability distribution is computed as follows
119885Dis = (120591Dis4 minus 1199051198774 )1205904 (24)
where 120591Dis4 represents the value of the L-Ck of the fitteddistribution 1199051198774 represents the weighted regional averageL-Ck and 1205904 represents standard deviation of the 1199051198774 whichis obtained from the simulation of the Kappa probabilitydistribution
If the computed value of 119885Dis is equal to zero the proba-bility distributionwill be themost suitable fit If the computed
6 Advances in Meteorology
Table 1 Statistics of annual extreme monthly precipitation for study region based on L-moments and PL-moments
Sites L-moments PL-momentsMean 119905 1199053 1199054 Mean 119905 1199053 1199054
value of 119885-statistic is less than 164 at 90 confidence level(ie |119885Dis| le 164) it will indicate that the distributionqualifies the goodness of fit criteria If there are more thanone distribution that qualify the criteria the most suitabledistribution has the minimum |119885Dis| value3 Estimation
The sitesrsquo information and statistic by using L-moments forthe present study are presented in Table 1 In Table 1 meanrepresents the first sample LPL-moments and 119905 1199053 and 1199054are the sample LPL-moments ratios of the L-CvPL-CvL-CsPL-Cs and L-CkPL-Ck respectively The lower levelcensoring threshold is selected from 10 to 23 Table 2expresses the feasible threshold values according to thepercentile technique along with Average Annual OccurrenceNumber (AAON) Jiang et al [41] and Yuguo [42] suggestedthat the optimal threshold can be obtained if the values ofAAON lie between 1 and 2 Table 2 shows that 90th percentileobservations are suitable for the optimum threshold selectionof most of areas in the present study We have 168 valuesin each station according to the above table Astore stationhas 32 threshold values due to which 16 values are beingcensored in 168 According to censored level 102 censoredlevel was selected Similarly 105 was selected for Balakotand Muzaffarabad According to this process maximumthreshold level 223 was selected for Gupis By using theabove process for each station 17 different censoring levelswere selected So we decide that the range from 10 to 23of censoring level should be kept for selecting thresholdvalues
31 Regional Frequency Analysis The following four steps areconsidered as prerequisite for frequency analysis Hoskingand Wallis [7 30]
(1) Data screening(2) Designing of the homogeneous region(3) Selection of an appropriate probability distribution(4) Parameters estimation of the appropriate probability
distribution
311 Data Screening In this study we use secondary dataafter carefully examining all locations for abnormalities andmissing observations Therefore we use 14 years of data forRFA that were obtained from seventeen locations
312 Discordance Test Table 3 shows 119863119894 result of (18) for17 locations of this study region It can be observed fromthe results of LPL-moments in Table 3 that the value of 119863-statistic varies from 007 to 244 If 119863119894 is greater than 3 thelocation is considered to be discordant from the rest of theregional data Hosking and Wallis [7] In this study regionno location is diagnosed as discordant (119863119894 ge 3) Thereforewe use all data for the development of the RFA based on L-moments and PL-moments
313 Regional Heterogeneity Measure The next step is theformation of the homogeneous region which is conven-tionally tougher and needs the higher number of subjectivejudgments The homogeneity conditions are defined as thelocations that have the same frequency distributions
Advances in Meteorology 7
Table 2 Precipitation threshold selection in GPA distribution for 17 stations
For the present study area realization of the Kappaprobability distribution is used to conduct the heterogeneitytest based on the L-moments and PL-moments
Number of simulations are 10000 for computing theheterogeneityWe computed the regional average L-momentsratios the regional PL-moments ratios and the correspond-ing parameter values of the fitted Kappa probability dis-tribution (see Table 4) Table 4 shows the results of theheterogeneity measure using L-moments and PL-momentsmethods It can be observed from Table 4 that the different
values for the119867-statistic are minus041 minus160 and minus289 based onL-moments andminus006minus18 andminus317 based onPL-momentsTherefore we concluded that by comparing these resultsand the heterogeneity conditions study region is acceptablyhomogeneous for L-moments and PL-moments No furthersubdivisions of the present study are necessary
32 Fitting Appropriate Probability Distribution After homo-geneity analysis of the study area a suitable probabilitydistribution is required for the RFAThe objective is not onlyto recognize a suitable probability distribution for RFA butalso to observe a probability distribution that will providerobust quantile estimate for each location and for the regionalgrowth cure List of candidate probability distributions forRFA is GLO GEV GPA and GNO
We plotted L-moments and PL-moments diagrams forpreliminary evaluation of the probability distribution for thestudy area
Figure 2 illustrates an analogy of the observed and hypo-thetical relationships of the probability distribution Figure 2shows that GLO distribution is not a suitable candidate forthe L-moments and PL-moments
Interestingly both analyses of the L-moments and the PL-moments diagram show that the sample average values areappropriately distinguished by the hypothetical L-momentsand PL-moments for GPA and GNO distributions
However it is hard to find a suitable probability distri-bution that fits most of the regional observed data Table 5shows the goodness of fit test results for candidate probabilitydistributions
Table 5 shows that GLO distribution failed the goodnessof fit test for both L-moments and for PL-moments methodsas the calculated value of the 119885-test for the GLO distribution
Advances in Meteorology 9
Table 4 Heterogeneity measures for the study region based on L-moments and PL-moments
Observed standard deviation of group L-Cv 0078 0077Simulated mean of standard deviation of group L-Cv 0084 0077Simulated standard deviation of standard deviation of group L-Cv 0015 0014Value of the heterogeneity measure1198671 minus0410 minus0060
Heterogeneity measure1198672Observed average of L-CvL-skewness distance 0105 0139Simulated mean of average L-CvL-skewness distance 0141 0139Simulated standard deviation of average L-CvL-skewness distance 0023 0022Value of the heterogeneity measure1198672 minus1600 minus1800
Heterogeneity measure1198673Observed average of L-skewnessL-kurtosis distance 0094 0088Simulated mean of average L-skewnessL-kurtosis distance 0175 0176Simulated standard deviation of average L-skewnessL-kurtosis distance 0028 0028Value of the heterogeneity measure1198673 minus2890 minus3170
Table 5 119885-test result for the goodness of fitMethod GLO GEV GNO GPAL-moments 196 138 061 minus042PL-moments 177 119 043 minus061Table 6 Regional parameters for the three candidate distributionsfor L-moments and PL-moments
Method Distribution Parameters120585 120572 119870L-moments
is larger than the critical value of 164 (at 90 confidencelevel)
It has been observed that the computed values of |119885Dis|are less than 164 (at 90 confidence level) namely GEVGNO andGPA distributions However GEV GNO andGPAdistributions are suitable for regional distribution based onL-moments and PL-moments methods and for obtaining thefuture estimates of the quantile
Further it can be noted that GPA distribution is suitablefor L-moments method (lowest critical |119885Dis| value) Simi-larly GNO distribution is suitable for PL-moments method(lowest critical value) Table 6 shows the estimates of theregional parameters for L-moments and PL-moments for thesuitable probability distribution
33 Estimation of the Quantiles The regional quantile esti-mates 119902(119865) with varying nonexceedance probability119865 for theGNO GEV and GPA distributions are presented in Table 7
based on L-moments and PL-moments Quantile functionis normally represented as 119902(sdot) for fitted regional frequencydistribution The quantile estimate at location 119894 is establishedby joining the estimate of 120583119894 and 119902(sdot)
Mathematical form of the quantile estimate with nonex-ceedance probability 119865 is
The regional growth curves for the GEV GNO and GPAdistributions are shown in Figure 3
Figure 3 shows the regional growth curves of each can-didate distribution for L-moments and PL-moments GEVGNO and GPA distributions are approximately identicalup until 100-year return period (119865 = 099) for both L-moments and PL-moments However afterward the growthcurves of the GPA distribution lie below the GEV and GNOdistributions
Therefore it is necessary to assess the performance ofregional quantile estimates
4 Accuracy of the Estimated Quantiles and theRegional Growth Curve
A Monte Carlo simulation is designed to assess accuracyof the regional quantile estimates that are obtained by theRFA We use logical L-moments algorithm that has beenreported by Hosking and Wallis [7] in Section 64 Thisalgorithm takes samples from a region that has comparablecharacteristics as of the actual region such as having the samerecord length same number of locations and the regionalL-moments ratios The area used for simulation shouldreport the plausible heterogeneity in the area and intersitedependency if exist (Hosking andWallis [7]) In the repeatedsampling procedure the quantile estimates are computed forthe different nonexceedance probabilities Suppose that at119898th repetition and location 119894 quantile estimate can bewritten
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Table 7 Regional quantile estimates with nonexceedance probability 119865Method Distribution 119865
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)
= 1119872 [[119872sum119898=1
(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12
(27)
Also for the estimated quantile the regional average bias andthe relative RMSE are
119861119877 (119865) = 1119873119873sum119894=1
119861119894 (119865)
119877119877 (119865) = 1119873119873sum119894=1
119877119894 (119865) (28)
We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval
119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)
Advances in Meteorology 11
Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le
119902 (119865)119871005 (119865) (30)
The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves
We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]
We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively
Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds
Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion
This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail
GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments
Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents
Appendix
The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows
PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by
119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
formula and unit-hydrograph procedure Subramanya [6]and it can also estimate the quantiles of extreme precipitation
Hosking and Wallis [7] showed that RFA method basedon L-moments is used to detect homogeneous regions toselect suitable regional frequency distribution and to predictextreme precipitation quantiles at region of interest
Whilst L-moment methodology is effective in estimatingparameters it may not valid for predicting high returnperiod events Wang [8] suggested that relatively small floodsmight create disturbance in the analysis To overcome suchsituation a censored sample can be used as by Cunnane [9]
Wang [8] proposed partial probability-weightedmoments (PPWMs) for fitting the probability distributionfunction to the censored sample Partial L-moments(PL-moments) are variants of the L-moments and similar toPPWMs PL-momentsmethod has used in fitting generalizedextreme value (GEV) distribution for censored flood samples(see Wang [8 10 11] and Bhattarai [12]) Bhattarai [12] foundthat censoring flood samples are nearly thirty percent ofbasic L-moments
Shabri et al [13] used Trim L-moments (TL-moments)for the RFA and compared its performance with L-momentsSaf [14] determined hydrologically homogeneous region andregional flood frequency estimates by using index-floodtechnique along with L-moments for the West Mediter-ranean River Turkey L-moments method is also used toassign a suitable regional distribution for the individualsubregions and to assess their homogeneity see Abolverdiand Khalili [15] Hussain and Pasha [16] suggested theregional flood frequency analysis based on L-momentsTheyused discordancy measure for data screening and used thefour-parameter Kappa distribution with 500 simulationsfor the heterogeneity analysis Zakaria et al [17] used thePL-moments technique and found another link for thehomogeneity analysis Shahzadi et al [18] showed that thegeneralized normal (GNO) distribution is suitable for theregional quantile estimation at maximum return period andthe GEV distribution for the overall regions at low returnperiod based on relative RMSE and relative absolute biasMost commonly used statistical distributions for high climatemodeling are as follows the logistic distribution with threeparameters lognormal distribution with three parametersLog Pearson type III GEV and generalized Pareto (GPA)(Coles [19] Katz et al [20] Abida and Ellouze [21] Feng et al[22] Yang et al [23] Villarini et al [24] Zakaria et al [17] Sheet al [25])Moreover GEV andGPAdistributions are suitableif data contains extremes values Over the years the GNOGEV generalized logistic (GLO) and GPA distributionshave been widely employed in the extreme value estimationof annual flood peaks In this study we aim to developRFA method based on L- and PL-moments approach Ourproposed method can be used at all levels of regional analysissuch as identification of homogeneous regions identificationand also testing of the suitable probability regional frequencydistribution based on 119885-statistic and the L- and PL-momentratio diagram and estimation of the flood quantile at siteof interest We use 17 sites of Northern areas and KhyberPakhtunkhwa as a case study to perform the analysis
We explain the methodology of the L-moments and PL-moments in Section 22 Section 23 shows the applicationof the RFA where we choose the appropriate distributionfor the regional analysis Section 33 provides the estimationof quantile for both the small and large return period Theresults of our simulations studywill be presented in Section 4
2 Material and Methods
21 Study Area and Data Sources The data was collectedfor this study from Karachi Data Processing Center throughPakistan Meteorological (PMD) Islamabad Monthly precip-itation data has been recorded from 1999 to 2012 There are17 meteorological stations of Northern areas and KhyberPakhtunkhwa Pakistan These stations are full precipitationregions that affect areas in Pakistan where water is essentialfor hydropower and flood plains Figure 1 shows the locationof the study area and geographic distribution of precipitationstations There is no missing value in this data set
22 Statistical Methods
221 The L-Moments Conventional moments method maybe used for estimating the parameters of probability distri-bution However this approach has some serious drawbacksRatio ofmoment estimators is biased and often assumption ofbeing normally distributed is violated Wallis et al [26] Fur-thermore it is sensitive to outliers Pearson [27]Therefore itis unreliable for skewed distributions
Hosking [28] proposed L-moments approach to over-come the above problems The L-moments may preciselydescribe the statistical properties of hydrological informa-tion and it can write as a linear function of the PWMs ThePWMs of order 119903 were properly described by Greenwood etal [29] as
where 119865 = 119865(119909) is the cumulative distribution function of arandom variable 119909 and 119903 is a nonnegative integer of the realnumber that is 119903 = 0 1 2 3 Therefore the first four L-moments which are the linear combinations of the PWMsare
1205821 = 12057301205822 = 21205731 minus 12057301205823 = 1205732 minus 61205731 + 12057301205824 = 201205733 minus 301205732 + 121205731 minus 1205730
Figure 1 Locations of Northern areas and Khyber Pakhtunkhwa and the meteorological stations (119909 axis indicates Longitude [E] 119910 axisindicates Latitude [N])
The L-moments have no units of measurement which arecalled the L-moments ratios The L-moments ratios pro-posed by Hosking [28] are computed as
120591 = 12058221205821
1205913 = 12058231205822
1205914 = 12058241205822 (5)
4 Advances in Meteorology
where 120591 represents the L-coefficient of variation (L-Cv)1205913 represents the L-coefficient of skewness (L-Cs) and 1205914represents the L-coefficient of kurtosis (L-Ck)
The arranged sample is given as 119909(1) le 119909(2) le 119909(3) sdot sdot sdot le119909(119899) Wang [8] stated that the statistic
119887119903 = 1119899119899sum119894=1
(119894 minus 1) (119894 minus 2) sdot sdot sdot (119894 minus 119903)(119899 minus 1) (119899 minus 2) sdot sdot sdot (119899 minus 119903)119909(119894) (6)
is an unbiased estimator of 120573119903So
1198971 = 11988701198972 = 21198871 minus 11988701198973 = 61198872 minus 61198871 + 11988701198974 = 201198873 minus 301198872 + 121198871 minus 1198870
(7)
where 1198971 1198972 1198973 and 1198974 are the first four L-moments of thesample And similarly
222 The Partial L-Moments Wang [8 10 11] introduced aconcept of partial probability-weighted moments (PPWMs)that will estimate the higher quantiles of flood flows Data canbe censored to the right tail or left tail
Initially PPWMswere to take out the smaller values fromthe process of distribution fitting because such values haveslight influence on the frequency analysis and are nuisance tothe fitting process
The left tail PPWMs are defined by Wang [8 11] as
120573lowast119903 = int11198650
119909 (119865) 119865119903119889119865 (9)
where 1198650 = 119865(1199090) which is the lower limit of the censoredobservations and 1199090 is the censoring threshold value
The PPWMs elongated form described by Wang [10] areto be given a censored sample as
120573119903 = 11 minus 119865119903+10 int11198650
119909 (119865) 119865119903119889119865 (10)
If the value of 1198650 is starting from the zero then the result ofPPWMswill be the same as the usual PWMs As 119909(1) le 119909(2) le119909(3) sdot sdot sdot le 119909(119899) is the arranged sample Wang [10] describes theunbiased estimator of 120573119903 as
119903 = 1(1 minus 119865119903+10 ) 119899119899sum119894=1
(119894 minus 1) (119894 minus 2) sdot sdot sdot (119894 minus 119903)(119899 minus 1) (119899 minus 2) sdot sdot sdot (119899 minus 119903)119909lowast(119894) (11)
where
119909lowast(119894) = 0 for 119909(119894) le 119909(0)119909lowast(119894) = 119909(119894) for 119909(119894) gt 119909(0) (12)
The censoring level 1198650 is the prior selection of the numberof censored sample data The procedure that determines thenumber of sample data points are to be censored
1198650 = 1198990119899 (13)
where 1198990 and 119899 are the lengths of sample which are to becensored and uncensored respectively Similarly 1199090 is thehighest value of the censored sample The first four PL-moments for the PPWMs are
1 = 12057302 = 2 1205731 minus 12057303 = 1205732 minus 6 1205731 + 12057304 = 20 1205733 minus 30 1205732 + 12 1205731 minus 1205730
(14)
Similarly the PL-moments ratios can be written as
120591 = 21
1205913 = 32
1205914 = 42
(15)
where 120591 1205913 and 1205914 denote the partial L-coefficient of variation(PL-Cv) partial L-coefficient of skewness (PL-Cs) and par-tial L-coefficient of kurtosis (PL-Ck) respectively Thereforethe first four sample PL-moments can be computed as
1198971 = 01198972 = 21 minus 01198973 = 62 minus 61 + 01198974 = 203 minus 302 + 121 minus 0
(16)
And the first four sample PL-moments ratios can be com-puted as
where 119905 1199053 and 1199054 represent the sample partial L-momentsratios of the PL-Cv PL-Cs and PL-Ck respectively Thederivation is L-moment and PL-moments are given inAppendix In the present study different level of censoringthreshold is selected
23 Regional Frequency Analysis Hosking and Wallis [7 30]identified the following four steps to explain the procedure ofthe RFA
(1) Data screening(2) Designing of the homogeneous region(3) Selection of an appropriate probability distribution(4) Parameters estimation of the appropriate probability
distribution
231 Data Screening We screened data anomalies beforeapplying any statistical analysis
232 Discordance Test Hosking and Wallis [7] suggested adiscordancy measure (119863119894) test that recognizes the locationswhere sample L-moments are marked contrarily from themost other locations Locations with the large flaws in thedata will be flagged as discordant
The discordancy test for a region containing119873 locationsfor site 119894 is proposed by Hosking and Wallis [7] as follows
119863119894 = 13119873 (119906119894 minus 119906)119879 119878minus1 (119906119894 minus 119906) 119894 = 1 2 119873 (18)
where 119906119894 is the vector containing the three sample L-momentsratios for the site 119894 expressed as
119906 is the average vector of 119906119894 for the overall region that is
119906 = 1119873119873sum119894=1
119906119894 (20)
and 119878 is the covariance matrix for the sample that can beexpressed as
119878 = 119873sum119894=1
(119906119894 minus 119906) (119906119894 minus 119906)119879 (21)
Broadly speaking a location or a site is considered to bediscordant from the whole region or group if the value of 119863119894is larger than the critical value
233 Heterogeneity Test The homogeneity measure (119867119895)identifies homogenous regions Hosking and Wallis [7] Itis also useful to tag the locations if they are plausible tohandle as a homogeneous region It estimates the amountof heterogeneity in the overall region The heterogeneity test(119867119895) is computed as
where 120583V119895 and 120590V119895 are representing the mean and standarddeviation of the simulated 119881119895 values Also
1198811 = 119873sum119894=1
119899119894 (119905(119894)2 minus 1199051198772 )sum119873119894=1 119899119894
12
1198812 =sum119873119894=1 [119899119894 (119905(119894)2 minus 1199051198772 )2 + (119905(119894)3 minus 1199051198773 )212]
sum119873119894=1 119899119894
1198813 =sum119873119894=1 [119899119894 (119905(119894)3 minus 1199051198773 )2 + (119905(119894)4 minus 1199051198774 )212]
sum119873119894=1 119899119894
(23)
Here 1199051198772 1199051198773 and 1199051198774 are region average L-moments or PL-moments ratios We assessed the heterogeneity of a region assuggested by Hosking and Wallis [7]
Region is acceptably homogeneous if119867 lt 1Region is possibly heterogeneous if 1 le 119867 lt 2Region is definitely heterogeneous if119867 ge 2
24 Selection of the Appropriate Probability DistributionHosking andWallis [7] proposed two approaches to select thedistribution that fitted best the data the L-moment ratios dia-gram and the 119885-test The L-moment ratios diagram is usingthe unbiased estimators Hosking [28] Stedinger et al [31]Vogel and Fennessey [32] and Hosking [33]The L-momentsratio diagram is a plot of the computed values L-Cs andthe observed values L-Ck of the distribution function Thecurves indicate the hypothetical connections between L-Csand L-Ck of the candidate distribution The L-moment ratiodiagrams have been proposed for discriminating between thecandidate probability distributions in describing the regionalinformation (Hosking [28] Stedinger et al [31] Hoskingand Wallis [7]) The L-moments ratio diagrams have beenused as a component of probability distribution process forregional information (Schaefer [34] Pearson [35] Vogel andFennessey [32] Vogel et al [36] Chow and Watt [37] OnOzand Bayazit [38] Vogel and Wilson [39] Peel et al [40])
Hosking and Wallis [7] suggested a measure to see howwell the L-Cs and L-Ck of the fitted probability distributionmatch the regional average L-Cs and L-Ck of the observedinformation
The measure goodness of fit for every single selectedprobability distribution is computed as follows
119885Dis = (120591Dis4 minus 1199051198774 )1205904 (24)
where 120591Dis4 represents the value of the L-Ck of the fitteddistribution 1199051198774 represents the weighted regional averageL-Ck and 1205904 represents standard deviation of the 1199051198774 whichis obtained from the simulation of the Kappa probabilitydistribution
If the computed value of 119885Dis is equal to zero the proba-bility distributionwill be themost suitable fit If the computed
6 Advances in Meteorology
Table 1 Statistics of annual extreme monthly precipitation for study region based on L-moments and PL-moments
Sites L-moments PL-momentsMean 119905 1199053 1199054 Mean 119905 1199053 1199054
value of 119885-statistic is less than 164 at 90 confidence level(ie |119885Dis| le 164) it will indicate that the distributionqualifies the goodness of fit criteria If there are more thanone distribution that qualify the criteria the most suitabledistribution has the minimum |119885Dis| value3 Estimation
The sitesrsquo information and statistic by using L-moments forthe present study are presented in Table 1 In Table 1 meanrepresents the first sample LPL-moments and 119905 1199053 and 1199054are the sample LPL-moments ratios of the L-CvPL-CvL-CsPL-Cs and L-CkPL-Ck respectively The lower levelcensoring threshold is selected from 10 to 23 Table 2expresses the feasible threshold values according to thepercentile technique along with Average Annual OccurrenceNumber (AAON) Jiang et al [41] and Yuguo [42] suggestedthat the optimal threshold can be obtained if the values ofAAON lie between 1 and 2 Table 2 shows that 90th percentileobservations are suitable for the optimum threshold selectionof most of areas in the present study We have 168 valuesin each station according to the above table Astore stationhas 32 threshold values due to which 16 values are beingcensored in 168 According to censored level 102 censoredlevel was selected Similarly 105 was selected for Balakotand Muzaffarabad According to this process maximumthreshold level 223 was selected for Gupis By using theabove process for each station 17 different censoring levelswere selected So we decide that the range from 10 to 23of censoring level should be kept for selecting thresholdvalues
31 Regional Frequency Analysis The following four steps areconsidered as prerequisite for frequency analysis Hoskingand Wallis [7 30]
(1) Data screening(2) Designing of the homogeneous region(3) Selection of an appropriate probability distribution(4) Parameters estimation of the appropriate probability
distribution
311 Data Screening In this study we use secondary dataafter carefully examining all locations for abnormalities andmissing observations Therefore we use 14 years of data forRFA that were obtained from seventeen locations
312 Discordance Test Table 3 shows 119863119894 result of (18) for17 locations of this study region It can be observed fromthe results of LPL-moments in Table 3 that the value of 119863-statistic varies from 007 to 244 If 119863119894 is greater than 3 thelocation is considered to be discordant from the rest of theregional data Hosking and Wallis [7] In this study regionno location is diagnosed as discordant (119863119894 ge 3) Thereforewe use all data for the development of the RFA based on L-moments and PL-moments
313 Regional Heterogeneity Measure The next step is theformation of the homogeneous region which is conven-tionally tougher and needs the higher number of subjectivejudgments The homogeneity conditions are defined as thelocations that have the same frequency distributions
Advances in Meteorology 7
Table 2 Precipitation threshold selection in GPA distribution for 17 stations
For the present study area realization of the Kappaprobability distribution is used to conduct the heterogeneitytest based on the L-moments and PL-moments
Number of simulations are 10000 for computing theheterogeneityWe computed the regional average L-momentsratios the regional PL-moments ratios and the correspond-ing parameter values of the fitted Kappa probability dis-tribution (see Table 4) Table 4 shows the results of theheterogeneity measure using L-moments and PL-momentsmethods It can be observed from Table 4 that the different
values for the119867-statistic are minus041 minus160 and minus289 based onL-moments andminus006minus18 andminus317 based onPL-momentsTherefore we concluded that by comparing these resultsand the heterogeneity conditions study region is acceptablyhomogeneous for L-moments and PL-moments No furthersubdivisions of the present study are necessary
32 Fitting Appropriate Probability Distribution After homo-geneity analysis of the study area a suitable probabilitydistribution is required for the RFAThe objective is not onlyto recognize a suitable probability distribution for RFA butalso to observe a probability distribution that will providerobust quantile estimate for each location and for the regionalgrowth cure List of candidate probability distributions forRFA is GLO GEV GPA and GNO
We plotted L-moments and PL-moments diagrams forpreliminary evaluation of the probability distribution for thestudy area
Figure 2 illustrates an analogy of the observed and hypo-thetical relationships of the probability distribution Figure 2shows that GLO distribution is not a suitable candidate forthe L-moments and PL-moments
Interestingly both analyses of the L-moments and the PL-moments diagram show that the sample average values areappropriately distinguished by the hypothetical L-momentsand PL-moments for GPA and GNO distributions
However it is hard to find a suitable probability distri-bution that fits most of the regional observed data Table 5shows the goodness of fit test results for candidate probabilitydistributions
Table 5 shows that GLO distribution failed the goodnessof fit test for both L-moments and for PL-moments methodsas the calculated value of the 119885-test for the GLO distribution
Advances in Meteorology 9
Table 4 Heterogeneity measures for the study region based on L-moments and PL-moments
Observed standard deviation of group L-Cv 0078 0077Simulated mean of standard deviation of group L-Cv 0084 0077Simulated standard deviation of standard deviation of group L-Cv 0015 0014Value of the heterogeneity measure1198671 minus0410 minus0060
Heterogeneity measure1198672Observed average of L-CvL-skewness distance 0105 0139Simulated mean of average L-CvL-skewness distance 0141 0139Simulated standard deviation of average L-CvL-skewness distance 0023 0022Value of the heterogeneity measure1198672 minus1600 minus1800
Heterogeneity measure1198673Observed average of L-skewnessL-kurtosis distance 0094 0088Simulated mean of average L-skewnessL-kurtosis distance 0175 0176Simulated standard deviation of average L-skewnessL-kurtosis distance 0028 0028Value of the heterogeneity measure1198673 minus2890 minus3170
Table 5 119885-test result for the goodness of fitMethod GLO GEV GNO GPAL-moments 196 138 061 minus042PL-moments 177 119 043 minus061Table 6 Regional parameters for the three candidate distributionsfor L-moments and PL-moments
Method Distribution Parameters120585 120572 119870L-moments
is larger than the critical value of 164 (at 90 confidencelevel)
It has been observed that the computed values of |119885Dis|are less than 164 (at 90 confidence level) namely GEVGNO andGPA distributions However GEV GNO andGPAdistributions are suitable for regional distribution based onL-moments and PL-moments methods and for obtaining thefuture estimates of the quantile
Further it can be noted that GPA distribution is suitablefor L-moments method (lowest critical |119885Dis| value) Simi-larly GNO distribution is suitable for PL-moments method(lowest critical value) Table 6 shows the estimates of theregional parameters for L-moments and PL-moments for thesuitable probability distribution
33 Estimation of the Quantiles The regional quantile esti-mates 119902(119865) with varying nonexceedance probability119865 for theGNO GEV and GPA distributions are presented in Table 7
based on L-moments and PL-moments Quantile functionis normally represented as 119902(sdot) for fitted regional frequencydistribution The quantile estimate at location 119894 is establishedby joining the estimate of 120583119894 and 119902(sdot)
Mathematical form of the quantile estimate with nonex-ceedance probability 119865 is
The regional growth curves for the GEV GNO and GPAdistributions are shown in Figure 3
Figure 3 shows the regional growth curves of each can-didate distribution for L-moments and PL-moments GEVGNO and GPA distributions are approximately identicalup until 100-year return period (119865 = 099) for both L-moments and PL-moments However afterward the growthcurves of the GPA distribution lie below the GEV and GNOdistributions
Therefore it is necessary to assess the performance ofregional quantile estimates
4 Accuracy of the Estimated Quantiles and theRegional Growth Curve
A Monte Carlo simulation is designed to assess accuracyof the regional quantile estimates that are obtained by theRFA We use logical L-moments algorithm that has beenreported by Hosking and Wallis [7] in Section 64 Thisalgorithm takes samples from a region that has comparablecharacteristics as of the actual region such as having the samerecord length same number of locations and the regionalL-moments ratios The area used for simulation shouldreport the plausible heterogeneity in the area and intersitedependency if exist (Hosking andWallis [7]) In the repeatedsampling procedure the quantile estimates are computed forthe different nonexceedance probabilities Suppose that at119898th repetition and location 119894 quantile estimate can bewritten
10 Advances in Meteorology
Table 7 Regional quantile estimates with nonexceedance probability 119865Method Distribution 119865
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)
= 1119872 [[119872sum119898=1
(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12
(27)
Also for the estimated quantile the regional average bias andthe relative RMSE are
119861119877 (119865) = 1119873119873sum119894=1
119861119894 (119865)
119877119877 (119865) = 1119873119873sum119894=1
119877119894 (119865) (28)
We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval
119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)
Advances in Meteorology 11
Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le
119902 (119865)119871005 (119865) (30)
The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves
We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]
We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively
Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds
Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion
This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail
GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments
Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents
Appendix
The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows
PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by
119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
Figure 1 Locations of Northern areas and Khyber Pakhtunkhwa and the meteorological stations (119909 axis indicates Longitude [E] 119910 axisindicates Latitude [N])
The L-moments have no units of measurement which arecalled the L-moments ratios The L-moments ratios pro-posed by Hosking [28] are computed as
120591 = 12058221205821
1205913 = 12058231205822
1205914 = 12058241205822 (5)
4 Advances in Meteorology
where 120591 represents the L-coefficient of variation (L-Cv)1205913 represents the L-coefficient of skewness (L-Cs) and 1205914represents the L-coefficient of kurtosis (L-Ck)
The arranged sample is given as 119909(1) le 119909(2) le 119909(3) sdot sdot sdot le119909(119899) Wang [8] stated that the statistic
119887119903 = 1119899119899sum119894=1
(119894 minus 1) (119894 minus 2) sdot sdot sdot (119894 minus 119903)(119899 minus 1) (119899 minus 2) sdot sdot sdot (119899 minus 119903)119909(119894) (6)
is an unbiased estimator of 120573119903So
1198971 = 11988701198972 = 21198871 minus 11988701198973 = 61198872 minus 61198871 + 11988701198974 = 201198873 minus 301198872 + 121198871 minus 1198870
(7)
where 1198971 1198972 1198973 and 1198974 are the first four L-moments of thesample And similarly
222 The Partial L-Moments Wang [8 10 11] introduced aconcept of partial probability-weighted moments (PPWMs)that will estimate the higher quantiles of flood flows Data canbe censored to the right tail or left tail
Initially PPWMswere to take out the smaller values fromthe process of distribution fitting because such values haveslight influence on the frequency analysis and are nuisance tothe fitting process
The left tail PPWMs are defined by Wang [8 11] as
120573lowast119903 = int11198650
119909 (119865) 119865119903119889119865 (9)
where 1198650 = 119865(1199090) which is the lower limit of the censoredobservations and 1199090 is the censoring threshold value
The PPWMs elongated form described by Wang [10] areto be given a censored sample as
120573119903 = 11 minus 119865119903+10 int11198650
119909 (119865) 119865119903119889119865 (10)
If the value of 1198650 is starting from the zero then the result ofPPWMswill be the same as the usual PWMs As 119909(1) le 119909(2) le119909(3) sdot sdot sdot le 119909(119899) is the arranged sample Wang [10] describes theunbiased estimator of 120573119903 as
119903 = 1(1 minus 119865119903+10 ) 119899119899sum119894=1
(119894 minus 1) (119894 minus 2) sdot sdot sdot (119894 minus 119903)(119899 minus 1) (119899 minus 2) sdot sdot sdot (119899 minus 119903)119909lowast(119894) (11)
where
119909lowast(119894) = 0 for 119909(119894) le 119909(0)119909lowast(119894) = 119909(119894) for 119909(119894) gt 119909(0) (12)
The censoring level 1198650 is the prior selection of the numberof censored sample data The procedure that determines thenumber of sample data points are to be censored
1198650 = 1198990119899 (13)
where 1198990 and 119899 are the lengths of sample which are to becensored and uncensored respectively Similarly 1199090 is thehighest value of the censored sample The first four PL-moments for the PPWMs are
1 = 12057302 = 2 1205731 minus 12057303 = 1205732 minus 6 1205731 + 12057304 = 20 1205733 minus 30 1205732 + 12 1205731 minus 1205730
(14)
Similarly the PL-moments ratios can be written as
120591 = 21
1205913 = 32
1205914 = 42
(15)
where 120591 1205913 and 1205914 denote the partial L-coefficient of variation(PL-Cv) partial L-coefficient of skewness (PL-Cs) and par-tial L-coefficient of kurtosis (PL-Ck) respectively Thereforethe first four sample PL-moments can be computed as
1198971 = 01198972 = 21 minus 01198973 = 62 minus 61 + 01198974 = 203 minus 302 + 121 minus 0
(16)
And the first four sample PL-moments ratios can be com-puted as
where 119905 1199053 and 1199054 represent the sample partial L-momentsratios of the PL-Cv PL-Cs and PL-Ck respectively Thederivation is L-moment and PL-moments are given inAppendix In the present study different level of censoringthreshold is selected
23 Regional Frequency Analysis Hosking and Wallis [7 30]identified the following four steps to explain the procedure ofthe RFA
(1) Data screening(2) Designing of the homogeneous region(3) Selection of an appropriate probability distribution(4) Parameters estimation of the appropriate probability
distribution
231 Data Screening We screened data anomalies beforeapplying any statistical analysis
232 Discordance Test Hosking and Wallis [7] suggested adiscordancy measure (119863119894) test that recognizes the locationswhere sample L-moments are marked contrarily from themost other locations Locations with the large flaws in thedata will be flagged as discordant
The discordancy test for a region containing119873 locationsfor site 119894 is proposed by Hosking and Wallis [7] as follows
119863119894 = 13119873 (119906119894 minus 119906)119879 119878minus1 (119906119894 minus 119906) 119894 = 1 2 119873 (18)
where 119906119894 is the vector containing the three sample L-momentsratios for the site 119894 expressed as
119906 is the average vector of 119906119894 for the overall region that is
119906 = 1119873119873sum119894=1
119906119894 (20)
and 119878 is the covariance matrix for the sample that can beexpressed as
119878 = 119873sum119894=1
(119906119894 minus 119906) (119906119894 minus 119906)119879 (21)
Broadly speaking a location or a site is considered to bediscordant from the whole region or group if the value of 119863119894is larger than the critical value
233 Heterogeneity Test The homogeneity measure (119867119895)identifies homogenous regions Hosking and Wallis [7] Itis also useful to tag the locations if they are plausible tohandle as a homogeneous region It estimates the amountof heterogeneity in the overall region The heterogeneity test(119867119895) is computed as
where 120583V119895 and 120590V119895 are representing the mean and standarddeviation of the simulated 119881119895 values Also
1198811 = 119873sum119894=1
119899119894 (119905(119894)2 minus 1199051198772 )sum119873119894=1 119899119894
12
1198812 =sum119873119894=1 [119899119894 (119905(119894)2 minus 1199051198772 )2 + (119905(119894)3 minus 1199051198773 )212]
sum119873119894=1 119899119894
1198813 =sum119873119894=1 [119899119894 (119905(119894)3 minus 1199051198773 )2 + (119905(119894)4 minus 1199051198774 )212]
sum119873119894=1 119899119894
(23)
Here 1199051198772 1199051198773 and 1199051198774 are region average L-moments or PL-moments ratios We assessed the heterogeneity of a region assuggested by Hosking and Wallis [7]
Region is acceptably homogeneous if119867 lt 1Region is possibly heterogeneous if 1 le 119867 lt 2Region is definitely heterogeneous if119867 ge 2
24 Selection of the Appropriate Probability DistributionHosking andWallis [7] proposed two approaches to select thedistribution that fitted best the data the L-moment ratios dia-gram and the 119885-test The L-moment ratios diagram is usingthe unbiased estimators Hosking [28] Stedinger et al [31]Vogel and Fennessey [32] and Hosking [33]The L-momentsratio diagram is a plot of the computed values L-Cs andthe observed values L-Ck of the distribution function Thecurves indicate the hypothetical connections between L-Csand L-Ck of the candidate distribution The L-moment ratiodiagrams have been proposed for discriminating between thecandidate probability distributions in describing the regionalinformation (Hosking [28] Stedinger et al [31] Hoskingand Wallis [7]) The L-moments ratio diagrams have beenused as a component of probability distribution process forregional information (Schaefer [34] Pearson [35] Vogel andFennessey [32] Vogel et al [36] Chow and Watt [37] OnOzand Bayazit [38] Vogel and Wilson [39] Peel et al [40])
Hosking and Wallis [7] suggested a measure to see howwell the L-Cs and L-Ck of the fitted probability distributionmatch the regional average L-Cs and L-Ck of the observedinformation
The measure goodness of fit for every single selectedprobability distribution is computed as follows
119885Dis = (120591Dis4 minus 1199051198774 )1205904 (24)
where 120591Dis4 represents the value of the L-Ck of the fitteddistribution 1199051198774 represents the weighted regional averageL-Ck and 1205904 represents standard deviation of the 1199051198774 whichis obtained from the simulation of the Kappa probabilitydistribution
If the computed value of 119885Dis is equal to zero the proba-bility distributionwill be themost suitable fit If the computed
6 Advances in Meteorology
Table 1 Statistics of annual extreme monthly precipitation for study region based on L-moments and PL-moments
Sites L-moments PL-momentsMean 119905 1199053 1199054 Mean 119905 1199053 1199054
value of 119885-statistic is less than 164 at 90 confidence level(ie |119885Dis| le 164) it will indicate that the distributionqualifies the goodness of fit criteria If there are more thanone distribution that qualify the criteria the most suitabledistribution has the minimum |119885Dis| value3 Estimation
The sitesrsquo information and statistic by using L-moments forthe present study are presented in Table 1 In Table 1 meanrepresents the first sample LPL-moments and 119905 1199053 and 1199054are the sample LPL-moments ratios of the L-CvPL-CvL-CsPL-Cs and L-CkPL-Ck respectively The lower levelcensoring threshold is selected from 10 to 23 Table 2expresses the feasible threshold values according to thepercentile technique along with Average Annual OccurrenceNumber (AAON) Jiang et al [41] and Yuguo [42] suggestedthat the optimal threshold can be obtained if the values ofAAON lie between 1 and 2 Table 2 shows that 90th percentileobservations are suitable for the optimum threshold selectionof most of areas in the present study We have 168 valuesin each station according to the above table Astore stationhas 32 threshold values due to which 16 values are beingcensored in 168 According to censored level 102 censoredlevel was selected Similarly 105 was selected for Balakotand Muzaffarabad According to this process maximumthreshold level 223 was selected for Gupis By using theabove process for each station 17 different censoring levelswere selected So we decide that the range from 10 to 23of censoring level should be kept for selecting thresholdvalues
31 Regional Frequency Analysis The following four steps areconsidered as prerequisite for frequency analysis Hoskingand Wallis [7 30]
(1) Data screening(2) Designing of the homogeneous region(3) Selection of an appropriate probability distribution(4) Parameters estimation of the appropriate probability
distribution
311 Data Screening In this study we use secondary dataafter carefully examining all locations for abnormalities andmissing observations Therefore we use 14 years of data forRFA that were obtained from seventeen locations
312 Discordance Test Table 3 shows 119863119894 result of (18) for17 locations of this study region It can be observed fromthe results of LPL-moments in Table 3 that the value of 119863-statistic varies from 007 to 244 If 119863119894 is greater than 3 thelocation is considered to be discordant from the rest of theregional data Hosking and Wallis [7] In this study regionno location is diagnosed as discordant (119863119894 ge 3) Thereforewe use all data for the development of the RFA based on L-moments and PL-moments
313 Regional Heterogeneity Measure The next step is theformation of the homogeneous region which is conven-tionally tougher and needs the higher number of subjectivejudgments The homogeneity conditions are defined as thelocations that have the same frequency distributions
Advances in Meteorology 7
Table 2 Precipitation threshold selection in GPA distribution for 17 stations
For the present study area realization of the Kappaprobability distribution is used to conduct the heterogeneitytest based on the L-moments and PL-moments
Number of simulations are 10000 for computing theheterogeneityWe computed the regional average L-momentsratios the regional PL-moments ratios and the correspond-ing parameter values of the fitted Kappa probability dis-tribution (see Table 4) Table 4 shows the results of theheterogeneity measure using L-moments and PL-momentsmethods It can be observed from Table 4 that the different
values for the119867-statistic are minus041 minus160 and minus289 based onL-moments andminus006minus18 andminus317 based onPL-momentsTherefore we concluded that by comparing these resultsand the heterogeneity conditions study region is acceptablyhomogeneous for L-moments and PL-moments No furthersubdivisions of the present study are necessary
32 Fitting Appropriate Probability Distribution After homo-geneity analysis of the study area a suitable probabilitydistribution is required for the RFAThe objective is not onlyto recognize a suitable probability distribution for RFA butalso to observe a probability distribution that will providerobust quantile estimate for each location and for the regionalgrowth cure List of candidate probability distributions forRFA is GLO GEV GPA and GNO
We plotted L-moments and PL-moments diagrams forpreliminary evaluation of the probability distribution for thestudy area
Figure 2 illustrates an analogy of the observed and hypo-thetical relationships of the probability distribution Figure 2shows that GLO distribution is not a suitable candidate forthe L-moments and PL-moments
Interestingly both analyses of the L-moments and the PL-moments diagram show that the sample average values areappropriately distinguished by the hypothetical L-momentsand PL-moments for GPA and GNO distributions
However it is hard to find a suitable probability distri-bution that fits most of the regional observed data Table 5shows the goodness of fit test results for candidate probabilitydistributions
Table 5 shows that GLO distribution failed the goodnessof fit test for both L-moments and for PL-moments methodsas the calculated value of the 119885-test for the GLO distribution
Advances in Meteorology 9
Table 4 Heterogeneity measures for the study region based on L-moments and PL-moments
Observed standard deviation of group L-Cv 0078 0077Simulated mean of standard deviation of group L-Cv 0084 0077Simulated standard deviation of standard deviation of group L-Cv 0015 0014Value of the heterogeneity measure1198671 minus0410 minus0060
Heterogeneity measure1198672Observed average of L-CvL-skewness distance 0105 0139Simulated mean of average L-CvL-skewness distance 0141 0139Simulated standard deviation of average L-CvL-skewness distance 0023 0022Value of the heterogeneity measure1198672 minus1600 minus1800
Heterogeneity measure1198673Observed average of L-skewnessL-kurtosis distance 0094 0088Simulated mean of average L-skewnessL-kurtosis distance 0175 0176Simulated standard deviation of average L-skewnessL-kurtosis distance 0028 0028Value of the heterogeneity measure1198673 minus2890 minus3170
Table 5 119885-test result for the goodness of fitMethod GLO GEV GNO GPAL-moments 196 138 061 minus042PL-moments 177 119 043 minus061Table 6 Regional parameters for the three candidate distributionsfor L-moments and PL-moments
Method Distribution Parameters120585 120572 119870L-moments
is larger than the critical value of 164 (at 90 confidencelevel)
It has been observed that the computed values of |119885Dis|are less than 164 (at 90 confidence level) namely GEVGNO andGPA distributions However GEV GNO andGPAdistributions are suitable for regional distribution based onL-moments and PL-moments methods and for obtaining thefuture estimates of the quantile
Further it can be noted that GPA distribution is suitablefor L-moments method (lowest critical |119885Dis| value) Simi-larly GNO distribution is suitable for PL-moments method(lowest critical value) Table 6 shows the estimates of theregional parameters for L-moments and PL-moments for thesuitable probability distribution
33 Estimation of the Quantiles The regional quantile esti-mates 119902(119865) with varying nonexceedance probability119865 for theGNO GEV and GPA distributions are presented in Table 7
based on L-moments and PL-moments Quantile functionis normally represented as 119902(sdot) for fitted regional frequencydistribution The quantile estimate at location 119894 is establishedby joining the estimate of 120583119894 and 119902(sdot)
Mathematical form of the quantile estimate with nonex-ceedance probability 119865 is
The regional growth curves for the GEV GNO and GPAdistributions are shown in Figure 3
Figure 3 shows the regional growth curves of each can-didate distribution for L-moments and PL-moments GEVGNO and GPA distributions are approximately identicalup until 100-year return period (119865 = 099) for both L-moments and PL-moments However afterward the growthcurves of the GPA distribution lie below the GEV and GNOdistributions
Therefore it is necessary to assess the performance ofregional quantile estimates
4 Accuracy of the Estimated Quantiles and theRegional Growth Curve
A Monte Carlo simulation is designed to assess accuracyof the regional quantile estimates that are obtained by theRFA We use logical L-moments algorithm that has beenreported by Hosking and Wallis [7] in Section 64 Thisalgorithm takes samples from a region that has comparablecharacteristics as of the actual region such as having the samerecord length same number of locations and the regionalL-moments ratios The area used for simulation shouldreport the plausible heterogeneity in the area and intersitedependency if exist (Hosking andWallis [7]) In the repeatedsampling procedure the quantile estimates are computed forthe different nonexceedance probabilities Suppose that at119898th repetition and location 119894 quantile estimate can bewritten
10 Advances in Meteorology
Table 7 Regional quantile estimates with nonexceedance probability 119865Method Distribution 119865
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)
= 1119872 [[119872sum119898=1
(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12
(27)
Also for the estimated quantile the regional average bias andthe relative RMSE are
119861119877 (119865) = 1119873119873sum119894=1
119861119894 (119865)
119877119877 (119865) = 1119873119873sum119894=1
119877119894 (119865) (28)
We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval
119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)
Advances in Meteorology 11
Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le
119902 (119865)119871005 (119865) (30)
The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves
We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]
We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively
Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds
Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion
This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail
GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments
Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents
Appendix
The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows
PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by
119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
where 120591 represents the L-coefficient of variation (L-Cv)1205913 represents the L-coefficient of skewness (L-Cs) and 1205914represents the L-coefficient of kurtosis (L-Ck)
The arranged sample is given as 119909(1) le 119909(2) le 119909(3) sdot sdot sdot le119909(119899) Wang [8] stated that the statistic
119887119903 = 1119899119899sum119894=1
(119894 minus 1) (119894 minus 2) sdot sdot sdot (119894 minus 119903)(119899 minus 1) (119899 minus 2) sdot sdot sdot (119899 minus 119903)119909(119894) (6)
is an unbiased estimator of 120573119903So
1198971 = 11988701198972 = 21198871 minus 11988701198973 = 61198872 minus 61198871 + 11988701198974 = 201198873 minus 301198872 + 121198871 minus 1198870
(7)
where 1198971 1198972 1198973 and 1198974 are the first four L-moments of thesample And similarly
222 The Partial L-Moments Wang [8 10 11] introduced aconcept of partial probability-weighted moments (PPWMs)that will estimate the higher quantiles of flood flows Data canbe censored to the right tail or left tail
Initially PPWMswere to take out the smaller values fromthe process of distribution fitting because such values haveslight influence on the frequency analysis and are nuisance tothe fitting process
The left tail PPWMs are defined by Wang [8 11] as
120573lowast119903 = int11198650
119909 (119865) 119865119903119889119865 (9)
where 1198650 = 119865(1199090) which is the lower limit of the censoredobservations and 1199090 is the censoring threshold value
The PPWMs elongated form described by Wang [10] areto be given a censored sample as
120573119903 = 11 minus 119865119903+10 int11198650
119909 (119865) 119865119903119889119865 (10)
If the value of 1198650 is starting from the zero then the result ofPPWMswill be the same as the usual PWMs As 119909(1) le 119909(2) le119909(3) sdot sdot sdot le 119909(119899) is the arranged sample Wang [10] describes theunbiased estimator of 120573119903 as
119903 = 1(1 minus 119865119903+10 ) 119899119899sum119894=1
(119894 minus 1) (119894 minus 2) sdot sdot sdot (119894 minus 119903)(119899 minus 1) (119899 minus 2) sdot sdot sdot (119899 minus 119903)119909lowast(119894) (11)
where
119909lowast(119894) = 0 for 119909(119894) le 119909(0)119909lowast(119894) = 119909(119894) for 119909(119894) gt 119909(0) (12)
The censoring level 1198650 is the prior selection of the numberof censored sample data The procedure that determines thenumber of sample data points are to be censored
1198650 = 1198990119899 (13)
where 1198990 and 119899 are the lengths of sample which are to becensored and uncensored respectively Similarly 1199090 is thehighest value of the censored sample The first four PL-moments for the PPWMs are
1 = 12057302 = 2 1205731 minus 12057303 = 1205732 minus 6 1205731 + 12057304 = 20 1205733 minus 30 1205732 + 12 1205731 minus 1205730
(14)
Similarly the PL-moments ratios can be written as
120591 = 21
1205913 = 32
1205914 = 42
(15)
where 120591 1205913 and 1205914 denote the partial L-coefficient of variation(PL-Cv) partial L-coefficient of skewness (PL-Cs) and par-tial L-coefficient of kurtosis (PL-Ck) respectively Thereforethe first four sample PL-moments can be computed as
1198971 = 01198972 = 21 minus 01198973 = 62 minus 61 + 01198974 = 203 minus 302 + 121 minus 0
(16)
And the first four sample PL-moments ratios can be com-puted as
where 119905 1199053 and 1199054 represent the sample partial L-momentsratios of the PL-Cv PL-Cs and PL-Ck respectively Thederivation is L-moment and PL-moments are given inAppendix In the present study different level of censoringthreshold is selected
23 Regional Frequency Analysis Hosking and Wallis [7 30]identified the following four steps to explain the procedure ofthe RFA
(1) Data screening(2) Designing of the homogeneous region(3) Selection of an appropriate probability distribution(4) Parameters estimation of the appropriate probability
distribution
231 Data Screening We screened data anomalies beforeapplying any statistical analysis
232 Discordance Test Hosking and Wallis [7] suggested adiscordancy measure (119863119894) test that recognizes the locationswhere sample L-moments are marked contrarily from themost other locations Locations with the large flaws in thedata will be flagged as discordant
The discordancy test for a region containing119873 locationsfor site 119894 is proposed by Hosking and Wallis [7] as follows
119863119894 = 13119873 (119906119894 minus 119906)119879 119878minus1 (119906119894 minus 119906) 119894 = 1 2 119873 (18)
where 119906119894 is the vector containing the three sample L-momentsratios for the site 119894 expressed as
119906 is the average vector of 119906119894 for the overall region that is
119906 = 1119873119873sum119894=1
119906119894 (20)
and 119878 is the covariance matrix for the sample that can beexpressed as
119878 = 119873sum119894=1
(119906119894 minus 119906) (119906119894 minus 119906)119879 (21)
Broadly speaking a location or a site is considered to bediscordant from the whole region or group if the value of 119863119894is larger than the critical value
233 Heterogeneity Test The homogeneity measure (119867119895)identifies homogenous regions Hosking and Wallis [7] Itis also useful to tag the locations if they are plausible tohandle as a homogeneous region It estimates the amountof heterogeneity in the overall region The heterogeneity test(119867119895) is computed as
where 120583V119895 and 120590V119895 are representing the mean and standarddeviation of the simulated 119881119895 values Also
1198811 = 119873sum119894=1
119899119894 (119905(119894)2 minus 1199051198772 )sum119873119894=1 119899119894
12
1198812 =sum119873119894=1 [119899119894 (119905(119894)2 minus 1199051198772 )2 + (119905(119894)3 minus 1199051198773 )212]
sum119873119894=1 119899119894
1198813 =sum119873119894=1 [119899119894 (119905(119894)3 minus 1199051198773 )2 + (119905(119894)4 minus 1199051198774 )212]
sum119873119894=1 119899119894
(23)
Here 1199051198772 1199051198773 and 1199051198774 are region average L-moments or PL-moments ratios We assessed the heterogeneity of a region assuggested by Hosking and Wallis [7]
Region is acceptably homogeneous if119867 lt 1Region is possibly heterogeneous if 1 le 119867 lt 2Region is definitely heterogeneous if119867 ge 2
24 Selection of the Appropriate Probability DistributionHosking andWallis [7] proposed two approaches to select thedistribution that fitted best the data the L-moment ratios dia-gram and the 119885-test The L-moment ratios diagram is usingthe unbiased estimators Hosking [28] Stedinger et al [31]Vogel and Fennessey [32] and Hosking [33]The L-momentsratio diagram is a plot of the computed values L-Cs andthe observed values L-Ck of the distribution function Thecurves indicate the hypothetical connections between L-Csand L-Ck of the candidate distribution The L-moment ratiodiagrams have been proposed for discriminating between thecandidate probability distributions in describing the regionalinformation (Hosking [28] Stedinger et al [31] Hoskingand Wallis [7]) The L-moments ratio diagrams have beenused as a component of probability distribution process forregional information (Schaefer [34] Pearson [35] Vogel andFennessey [32] Vogel et al [36] Chow and Watt [37] OnOzand Bayazit [38] Vogel and Wilson [39] Peel et al [40])
Hosking and Wallis [7] suggested a measure to see howwell the L-Cs and L-Ck of the fitted probability distributionmatch the regional average L-Cs and L-Ck of the observedinformation
The measure goodness of fit for every single selectedprobability distribution is computed as follows
119885Dis = (120591Dis4 minus 1199051198774 )1205904 (24)
where 120591Dis4 represents the value of the L-Ck of the fitteddistribution 1199051198774 represents the weighted regional averageL-Ck and 1205904 represents standard deviation of the 1199051198774 whichis obtained from the simulation of the Kappa probabilitydistribution
If the computed value of 119885Dis is equal to zero the proba-bility distributionwill be themost suitable fit If the computed
6 Advances in Meteorology
Table 1 Statistics of annual extreme monthly precipitation for study region based on L-moments and PL-moments
Sites L-moments PL-momentsMean 119905 1199053 1199054 Mean 119905 1199053 1199054
value of 119885-statistic is less than 164 at 90 confidence level(ie |119885Dis| le 164) it will indicate that the distributionqualifies the goodness of fit criteria If there are more thanone distribution that qualify the criteria the most suitabledistribution has the minimum |119885Dis| value3 Estimation
The sitesrsquo information and statistic by using L-moments forthe present study are presented in Table 1 In Table 1 meanrepresents the first sample LPL-moments and 119905 1199053 and 1199054are the sample LPL-moments ratios of the L-CvPL-CvL-CsPL-Cs and L-CkPL-Ck respectively The lower levelcensoring threshold is selected from 10 to 23 Table 2expresses the feasible threshold values according to thepercentile technique along with Average Annual OccurrenceNumber (AAON) Jiang et al [41] and Yuguo [42] suggestedthat the optimal threshold can be obtained if the values ofAAON lie between 1 and 2 Table 2 shows that 90th percentileobservations are suitable for the optimum threshold selectionof most of areas in the present study We have 168 valuesin each station according to the above table Astore stationhas 32 threshold values due to which 16 values are beingcensored in 168 According to censored level 102 censoredlevel was selected Similarly 105 was selected for Balakotand Muzaffarabad According to this process maximumthreshold level 223 was selected for Gupis By using theabove process for each station 17 different censoring levelswere selected So we decide that the range from 10 to 23of censoring level should be kept for selecting thresholdvalues
31 Regional Frequency Analysis The following four steps areconsidered as prerequisite for frequency analysis Hoskingand Wallis [7 30]
(1) Data screening(2) Designing of the homogeneous region(3) Selection of an appropriate probability distribution(4) Parameters estimation of the appropriate probability
distribution
311 Data Screening In this study we use secondary dataafter carefully examining all locations for abnormalities andmissing observations Therefore we use 14 years of data forRFA that were obtained from seventeen locations
312 Discordance Test Table 3 shows 119863119894 result of (18) for17 locations of this study region It can be observed fromthe results of LPL-moments in Table 3 that the value of 119863-statistic varies from 007 to 244 If 119863119894 is greater than 3 thelocation is considered to be discordant from the rest of theregional data Hosking and Wallis [7] In this study regionno location is diagnosed as discordant (119863119894 ge 3) Thereforewe use all data for the development of the RFA based on L-moments and PL-moments
313 Regional Heterogeneity Measure The next step is theformation of the homogeneous region which is conven-tionally tougher and needs the higher number of subjectivejudgments The homogeneity conditions are defined as thelocations that have the same frequency distributions
Advances in Meteorology 7
Table 2 Precipitation threshold selection in GPA distribution for 17 stations
For the present study area realization of the Kappaprobability distribution is used to conduct the heterogeneitytest based on the L-moments and PL-moments
Number of simulations are 10000 for computing theheterogeneityWe computed the regional average L-momentsratios the regional PL-moments ratios and the correspond-ing parameter values of the fitted Kappa probability dis-tribution (see Table 4) Table 4 shows the results of theheterogeneity measure using L-moments and PL-momentsmethods It can be observed from Table 4 that the different
values for the119867-statistic are minus041 minus160 and minus289 based onL-moments andminus006minus18 andminus317 based onPL-momentsTherefore we concluded that by comparing these resultsand the heterogeneity conditions study region is acceptablyhomogeneous for L-moments and PL-moments No furthersubdivisions of the present study are necessary
32 Fitting Appropriate Probability Distribution After homo-geneity analysis of the study area a suitable probabilitydistribution is required for the RFAThe objective is not onlyto recognize a suitable probability distribution for RFA butalso to observe a probability distribution that will providerobust quantile estimate for each location and for the regionalgrowth cure List of candidate probability distributions forRFA is GLO GEV GPA and GNO
We plotted L-moments and PL-moments diagrams forpreliminary evaluation of the probability distribution for thestudy area
Figure 2 illustrates an analogy of the observed and hypo-thetical relationships of the probability distribution Figure 2shows that GLO distribution is not a suitable candidate forthe L-moments and PL-moments
Interestingly both analyses of the L-moments and the PL-moments diagram show that the sample average values areappropriately distinguished by the hypothetical L-momentsand PL-moments for GPA and GNO distributions
However it is hard to find a suitable probability distri-bution that fits most of the regional observed data Table 5shows the goodness of fit test results for candidate probabilitydistributions
Table 5 shows that GLO distribution failed the goodnessof fit test for both L-moments and for PL-moments methodsas the calculated value of the 119885-test for the GLO distribution
Advances in Meteorology 9
Table 4 Heterogeneity measures for the study region based on L-moments and PL-moments
Observed standard deviation of group L-Cv 0078 0077Simulated mean of standard deviation of group L-Cv 0084 0077Simulated standard deviation of standard deviation of group L-Cv 0015 0014Value of the heterogeneity measure1198671 minus0410 minus0060
Heterogeneity measure1198672Observed average of L-CvL-skewness distance 0105 0139Simulated mean of average L-CvL-skewness distance 0141 0139Simulated standard deviation of average L-CvL-skewness distance 0023 0022Value of the heterogeneity measure1198672 minus1600 minus1800
Heterogeneity measure1198673Observed average of L-skewnessL-kurtosis distance 0094 0088Simulated mean of average L-skewnessL-kurtosis distance 0175 0176Simulated standard deviation of average L-skewnessL-kurtosis distance 0028 0028Value of the heterogeneity measure1198673 minus2890 minus3170
Table 5 119885-test result for the goodness of fitMethod GLO GEV GNO GPAL-moments 196 138 061 minus042PL-moments 177 119 043 minus061Table 6 Regional parameters for the three candidate distributionsfor L-moments and PL-moments
Method Distribution Parameters120585 120572 119870L-moments
is larger than the critical value of 164 (at 90 confidencelevel)
It has been observed that the computed values of |119885Dis|are less than 164 (at 90 confidence level) namely GEVGNO andGPA distributions However GEV GNO andGPAdistributions are suitable for regional distribution based onL-moments and PL-moments methods and for obtaining thefuture estimates of the quantile
Further it can be noted that GPA distribution is suitablefor L-moments method (lowest critical |119885Dis| value) Simi-larly GNO distribution is suitable for PL-moments method(lowest critical value) Table 6 shows the estimates of theregional parameters for L-moments and PL-moments for thesuitable probability distribution
33 Estimation of the Quantiles The regional quantile esti-mates 119902(119865) with varying nonexceedance probability119865 for theGNO GEV and GPA distributions are presented in Table 7
based on L-moments and PL-moments Quantile functionis normally represented as 119902(sdot) for fitted regional frequencydistribution The quantile estimate at location 119894 is establishedby joining the estimate of 120583119894 and 119902(sdot)
Mathematical form of the quantile estimate with nonex-ceedance probability 119865 is
The regional growth curves for the GEV GNO and GPAdistributions are shown in Figure 3
Figure 3 shows the regional growth curves of each can-didate distribution for L-moments and PL-moments GEVGNO and GPA distributions are approximately identicalup until 100-year return period (119865 = 099) for both L-moments and PL-moments However afterward the growthcurves of the GPA distribution lie below the GEV and GNOdistributions
Therefore it is necessary to assess the performance ofregional quantile estimates
4 Accuracy of the Estimated Quantiles and theRegional Growth Curve
A Monte Carlo simulation is designed to assess accuracyof the regional quantile estimates that are obtained by theRFA We use logical L-moments algorithm that has beenreported by Hosking and Wallis [7] in Section 64 Thisalgorithm takes samples from a region that has comparablecharacteristics as of the actual region such as having the samerecord length same number of locations and the regionalL-moments ratios The area used for simulation shouldreport the plausible heterogeneity in the area and intersitedependency if exist (Hosking andWallis [7]) In the repeatedsampling procedure the quantile estimates are computed forthe different nonexceedance probabilities Suppose that at119898th repetition and location 119894 quantile estimate can bewritten
10 Advances in Meteorology
Table 7 Regional quantile estimates with nonexceedance probability 119865Method Distribution 119865
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)
= 1119872 [[119872sum119898=1
(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12
(27)
Also for the estimated quantile the regional average bias andthe relative RMSE are
119861119877 (119865) = 1119873119873sum119894=1
119861119894 (119865)
119877119877 (119865) = 1119873119873sum119894=1
119877119894 (119865) (28)
We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval
119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)
Advances in Meteorology 11
Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le
119902 (119865)119871005 (119865) (30)
The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves
We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]
We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively
Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds
Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion
This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail
GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments
Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents
Appendix
The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows
PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by
119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
where 119905 1199053 and 1199054 represent the sample partial L-momentsratios of the PL-Cv PL-Cs and PL-Ck respectively Thederivation is L-moment and PL-moments are given inAppendix In the present study different level of censoringthreshold is selected
23 Regional Frequency Analysis Hosking and Wallis [7 30]identified the following four steps to explain the procedure ofthe RFA
(1) Data screening(2) Designing of the homogeneous region(3) Selection of an appropriate probability distribution(4) Parameters estimation of the appropriate probability
distribution
231 Data Screening We screened data anomalies beforeapplying any statistical analysis
232 Discordance Test Hosking and Wallis [7] suggested adiscordancy measure (119863119894) test that recognizes the locationswhere sample L-moments are marked contrarily from themost other locations Locations with the large flaws in thedata will be flagged as discordant
The discordancy test for a region containing119873 locationsfor site 119894 is proposed by Hosking and Wallis [7] as follows
119863119894 = 13119873 (119906119894 minus 119906)119879 119878minus1 (119906119894 minus 119906) 119894 = 1 2 119873 (18)
where 119906119894 is the vector containing the three sample L-momentsratios for the site 119894 expressed as
119906 is the average vector of 119906119894 for the overall region that is
119906 = 1119873119873sum119894=1
119906119894 (20)
and 119878 is the covariance matrix for the sample that can beexpressed as
119878 = 119873sum119894=1
(119906119894 minus 119906) (119906119894 minus 119906)119879 (21)
Broadly speaking a location or a site is considered to bediscordant from the whole region or group if the value of 119863119894is larger than the critical value
233 Heterogeneity Test The homogeneity measure (119867119895)identifies homogenous regions Hosking and Wallis [7] Itis also useful to tag the locations if they are plausible tohandle as a homogeneous region It estimates the amountof heterogeneity in the overall region The heterogeneity test(119867119895) is computed as
where 120583V119895 and 120590V119895 are representing the mean and standarddeviation of the simulated 119881119895 values Also
1198811 = 119873sum119894=1
119899119894 (119905(119894)2 minus 1199051198772 )sum119873119894=1 119899119894
12
1198812 =sum119873119894=1 [119899119894 (119905(119894)2 minus 1199051198772 )2 + (119905(119894)3 minus 1199051198773 )212]
sum119873119894=1 119899119894
1198813 =sum119873119894=1 [119899119894 (119905(119894)3 minus 1199051198773 )2 + (119905(119894)4 minus 1199051198774 )212]
sum119873119894=1 119899119894
(23)
Here 1199051198772 1199051198773 and 1199051198774 are region average L-moments or PL-moments ratios We assessed the heterogeneity of a region assuggested by Hosking and Wallis [7]
Region is acceptably homogeneous if119867 lt 1Region is possibly heterogeneous if 1 le 119867 lt 2Region is definitely heterogeneous if119867 ge 2
24 Selection of the Appropriate Probability DistributionHosking andWallis [7] proposed two approaches to select thedistribution that fitted best the data the L-moment ratios dia-gram and the 119885-test The L-moment ratios diagram is usingthe unbiased estimators Hosking [28] Stedinger et al [31]Vogel and Fennessey [32] and Hosking [33]The L-momentsratio diagram is a plot of the computed values L-Cs andthe observed values L-Ck of the distribution function Thecurves indicate the hypothetical connections between L-Csand L-Ck of the candidate distribution The L-moment ratiodiagrams have been proposed for discriminating between thecandidate probability distributions in describing the regionalinformation (Hosking [28] Stedinger et al [31] Hoskingand Wallis [7]) The L-moments ratio diagrams have beenused as a component of probability distribution process forregional information (Schaefer [34] Pearson [35] Vogel andFennessey [32] Vogel et al [36] Chow and Watt [37] OnOzand Bayazit [38] Vogel and Wilson [39] Peel et al [40])
Hosking and Wallis [7] suggested a measure to see howwell the L-Cs and L-Ck of the fitted probability distributionmatch the regional average L-Cs and L-Ck of the observedinformation
The measure goodness of fit for every single selectedprobability distribution is computed as follows
119885Dis = (120591Dis4 minus 1199051198774 )1205904 (24)
where 120591Dis4 represents the value of the L-Ck of the fitteddistribution 1199051198774 represents the weighted regional averageL-Ck and 1205904 represents standard deviation of the 1199051198774 whichis obtained from the simulation of the Kappa probabilitydistribution
If the computed value of 119885Dis is equal to zero the proba-bility distributionwill be themost suitable fit If the computed
6 Advances in Meteorology
Table 1 Statistics of annual extreme monthly precipitation for study region based on L-moments and PL-moments
Sites L-moments PL-momentsMean 119905 1199053 1199054 Mean 119905 1199053 1199054
value of 119885-statistic is less than 164 at 90 confidence level(ie |119885Dis| le 164) it will indicate that the distributionqualifies the goodness of fit criteria If there are more thanone distribution that qualify the criteria the most suitabledistribution has the minimum |119885Dis| value3 Estimation
The sitesrsquo information and statistic by using L-moments forthe present study are presented in Table 1 In Table 1 meanrepresents the first sample LPL-moments and 119905 1199053 and 1199054are the sample LPL-moments ratios of the L-CvPL-CvL-CsPL-Cs and L-CkPL-Ck respectively The lower levelcensoring threshold is selected from 10 to 23 Table 2expresses the feasible threshold values according to thepercentile technique along with Average Annual OccurrenceNumber (AAON) Jiang et al [41] and Yuguo [42] suggestedthat the optimal threshold can be obtained if the values ofAAON lie between 1 and 2 Table 2 shows that 90th percentileobservations are suitable for the optimum threshold selectionof most of areas in the present study We have 168 valuesin each station according to the above table Astore stationhas 32 threshold values due to which 16 values are beingcensored in 168 According to censored level 102 censoredlevel was selected Similarly 105 was selected for Balakotand Muzaffarabad According to this process maximumthreshold level 223 was selected for Gupis By using theabove process for each station 17 different censoring levelswere selected So we decide that the range from 10 to 23of censoring level should be kept for selecting thresholdvalues
31 Regional Frequency Analysis The following four steps areconsidered as prerequisite for frequency analysis Hoskingand Wallis [7 30]
(1) Data screening(2) Designing of the homogeneous region(3) Selection of an appropriate probability distribution(4) Parameters estimation of the appropriate probability
distribution
311 Data Screening In this study we use secondary dataafter carefully examining all locations for abnormalities andmissing observations Therefore we use 14 years of data forRFA that were obtained from seventeen locations
312 Discordance Test Table 3 shows 119863119894 result of (18) for17 locations of this study region It can be observed fromthe results of LPL-moments in Table 3 that the value of 119863-statistic varies from 007 to 244 If 119863119894 is greater than 3 thelocation is considered to be discordant from the rest of theregional data Hosking and Wallis [7] In this study regionno location is diagnosed as discordant (119863119894 ge 3) Thereforewe use all data for the development of the RFA based on L-moments and PL-moments
313 Regional Heterogeneity Measure The next step is theformation of the homogeneous region which is conven-tionally tougher and needs the higher number of subjectivejudgments The homogeneity conditions are defined as thelocations that have the same frequency distributions
Advances in Meteorology 7
Table 2 Precipitation threshold selection in GPA distribution for 17 stations
For the present study area realization of the Kappaprobability distribution is used to conduct the heterogeneitytest based on the L-moments and PL-moments
Number of simulations are 10000 for computing theheterogeneityWe computed the regional average L-momentsratios the regional PL-moments ratios and the correspond-ing parameter values of the fitted Kappa probability dis-tribution (see Table 4) Table 4 shows the results of theheterogeneity measure using L-moments and PL-momentsmethods It can be observed from Table 4 that the different
values for the119867-statistic are minus041 minus160 and minus289 based onL-moments andminus006minus18 andminus317 based onPL-momentsTherefore we concluded that by comparing these resultsand the heterogeneity conditions study region is acceptablyhomogeneous for L-moments and PL-moments No furthersubdivisions of the present study are necessary
32 Fitting Appropriate Probability Distribution After homo-geneity analysis of the study area a suitable probabilitydistribution is required for the RFAThe objective is not onlyto recognize a suitable probability distribution for RFA butalso to observe a probability distribution that will providerobust quantile estimate for each location and for the regionalgrowth cure List of candidate probability distributions forRFA is GLO GEV GPA and GNO
We plotted L-moments and PL-moments diagrams forpreliminary evaluation of the probability distribution for thestudy area
Figure 2 illustrates an analogy of the observed and hypo-thetical relationships of the probability distribution Figure 2shows that GLO distribution is not a suitable candidate forthe L-moments and PL-moments
Interestingly both analyses of the L-moments and the PL-moments diagram show that the sample average values areappropriately distinguished by the hypothetical L-momentsand PL-moments for GPA and GNO distributions
However it is hard to find a suitable probability distri-bution that fits most of the regional observed data Table 5shows the goodness of fit test results for candidate probabilitydistributions
Table 5 shows that GLO distribution failed the goodnessof fit test for both L-moments and for PL-moments methodsas the calculated value of the 119885-test for the GLO distribution
Advances in Meteorology 9
Table 4 Heterogeneity measures for the study region based on L-moments and PL-moments
Observed standard deviation of group L-Cv 0078 0077Simulated mean of standard deviation of group L-Cv 0084 0077Simulated standard deviation of standard deviation of group L-Cv 0015 0014Value of the heterogeneity measure1198671 minus0410 minus0060
Heterogeneity measure1198672Observed average of L-CvL-skewness distance 0105 0139Simulated mean of average L-CvL-skewness distance 0141 0139Simulated standard deviation of average L-CvL-skewness distance 0023 0022Value of the heterogeneity measure1198672 minus1600 minus1800
Heterogeneity measure1198673Observed average of L-skewnessL-kurtosis distance 0094 0088Simulated mean of average L-skewnessL-kurtosis distance 0175 0176Simulated standard deviation of average L-skewnessL-kurtosis distance 0028 0028Value of the heterogeneity measure1198673 minus2890 minus3170
Table 5 119885-test result for the goodness of fitMethod GLO GEV GNO GPAL-moments 196 138 061 minus042PL-moments 177 119 043 minus061Table 6 Regional parameters for the three candidate distributionsfor L-moments and PL-moments
Method Distribution Parameters120585 120572 119870L-moments
is larger than the critical value of 164 (at 90 confidencelevel)
It has been observed that the computed values of |119885Dis|are less than 164 (at 90 confidence level) namely GEVGNO andGPA distributions However GEV GNO andGPAdistributions are suitable for regional distribution based onL-moments and PL-moments methods and for obtaining thefuture estimates of the quantile
Further it can be noted that GPA distribution is suitablefor L-moments method (lowest critical |119885Dis| value) Simi-larly GNO distribution is suitable for PL-moments method(lowest critical value) Table 6 shows the estimates of theregional parameters for L-moments and PL-moments for thesuitable probability distribution
33 Estimation of the Quantiles The regional quantile esti-mates 119902(119865) with varying nonexceedance probability119865 for theGNO GEV and GPA distributions are presented in Table 7
based on L-moments and PL-moments Quantile functionis normally represented as 119902(sdot) for fitted regional frequencydistribution The quantile estimate at location 119894 is establishedby joining the estimate of 120583119894 and 119902(sdot)
Mathematical form of the quantile estimate with nonex-ceedance probability 119865 is
The regional growth curves for the GEV GNO and GPAdistributions are shown in Figure 3
Figure 3 shows the regional growth curves of each can-didate distribution for L-moments and PL-moments GEVGNO and GPA distributions are approximately identicalup until 100-year return period (119865 = 099) for both L-moments and PL-moments However afterward the growthcurves of the GPA distribution lie below the GEV and GNOdistributions
Therefore it is necessary to assess the performance ofregional quantile estimates
4 Accuracy of the Estimated Quantiles and theRegional Growth Curve
A Monte Carlo simulation is designed to assess accuracyof the regional quantile estimates that are obtained by theRFA We use logical L-moments algorithm that has beenreported by Hosking and Wallis [7] in Section 64 Thisalgorithm takes samples from a region that has comparablecharacteristics as of the actual region such as having the samerecord length same number of locations and the regionalL-moments ratios The area used for simulation shouldreport the plausible heterogeneity in the area and intersitedependency if exist (Hosking andWallis [7]) In the repeatedsampling procedure the quantile estimates are computed forthe different nonexceedance probabilities Suppose that at119898th repetition and location 119894 quantile estimate can bewritten
10 Advances in Meteorology
Table 7 Regional quantile estimates with nonexceedance probability 119865Method Distribution 119865
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)
= 1119872 [[119872sum119898=1
(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12
(27)
Also for the estimated quantile the regional average bias andthe relative RMSE are
119861119877 (119865) = 1119873119873sum119894=1
119861119894 (119865)
119877119877 (119865) = 1119873119873sum119894=1
119877119894 (119865) (28)
We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval
119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)
Advances in Meteorology 11
Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le
119902 (119865)119871005 (119865) (30)
The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves
We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]
We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively
Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds
Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion
This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail
GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments
Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents
Appendix
The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows
PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by
119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
value of 119885-statistic is less than 164 at 90 confidence level(ie |119885Dis| le 164) it will indicate that the distributionqualifies the goodness of fit criteria If there are more thanone distribution that qualify the criteria the most suitabledistribution has the minimum |119885Dis| value3 Estimation
The sitesrsquo information and statistic by using L-moments forthe present study are presented in Table 1 In Table 1 meanrepresents the first sample LPL-moments and 119905 1199053 and 1199054are the sample LPL-moments ratios of the L-CvPL-CvL-CsPL-Cs and L-CkPL-Ck respectively The lower levelcensoring threshold is selected from 10 to 23 Table 2expresses the feasible threshold values according to thepercentile technique along with Average Annual OccurrenceNumber (AAON) Jiang et al [41] and Yuguo [42] suggestedthat the optimal threshold can be obtained if the values ofAAON lie between 1 and 2 Table 2 shows that 90th percentileobservations are suitable for the optimum threshold selectionof most of areas in the present study We have 168 valuesin each station according to the above table Astore stationhas 32 threshold values due to which 16 values are beingcensored in 168 According to censored level 102 censoredlevel was selected Similarly 105 was selected for Balakotand Muzaffarabad According to this process maximumthreshold level 223 was selected for Gupis By using theabove process for each station 17 different censoring levelswere selected So we decide that the range from 10 to 23of censoring level should be kept for selecting thresholdvalues
31 Regional Frequency Analysis The following four steps areconsidered as prerequisite for frequency analysis Hoskingand Wallis [7 30]
(1) Data screening(2) Designing of the homogeneous region(3) Selection of an appropriate probability distribution(4) Parameters estimation of the appropriate probability
distribution
311 Data Screening In this study we use secondary dataafter carefully examining all locations for abnormalities andmissing observations Therefore we use 14 years of data forRFA that were obtained from seventeen locations
312 Discordance Test Table 3 shows 119863119894 result of (18) for17 locations of this study region It can be observed fromthe results of LPL-moments in Table 3 that the value of 119863-statistic varies from 007 to 244 If 119863119894 is greater than 3 thelocation is considered to be discordant from the rest of theregional data Hosking and Wallis [7] In this study regionno location is diagnosed as discordant (119863119894 ge 3) Thereforewe use all data for the development of the RFA based on L-moments and PL-moments
313 Regional Heterogeneity Measure The next step is theformation of the homogeneous region which is conven-tionally tougher and needs the higher number of subjectivejudgments The homogeneity conditions are defined as thelocations that have the same frequency distributions
Advances in Meteorology 7
Table 2 Precipitation threshold selection in GPA distribution for 17 stations
For the present study area realization of the Kappaprobability distribution is used to conduct the heterogeneitytest based on the L-moments and PL-moments
Number of simulations are 10000 for computing theheterogeneityWe computed the regional average L-momentsratios the regional PL-moments ratios and the correspond-ing parameter values of the fitted Kappa probability dis-tribution (see Table 4) Table 4 shows the results of theheterogeneity measure using L-moments and PL-momentsmethods It can be observed from Table 4 that the different
values for the119867-statistic are minus041 minus160 and minus289 based onL-moments andminus006minus18 andminus317 based onPL-momentsTherefore we concluded that by comparing these resultsand the heterogeneity conditions study region is acceptablyhomogeneous for L-moments and PL-moments No furthersubdivisions of the present study are necessary
32 Fitting Appropriate Probability Distribution After homo-geneity analysis of the study area a suitable probabilitydistribution is required for the RFAThe objective is not onlyto recognize a suitable probability distribution for RFA butalso to observe a probability distribution that will providerobust quantile estimate for each location and for the regionalgrowth cure List of candidate probability distributions forRFA is GLO GEV GPA and GNO
We plotted L-moments and PL-moments diagrams forpreliminary evaluation of the probability distribution for thestudy area
Figure 2 illustrates an analogy of the observed and hypo-thetical relationships of the probability distribution Figure 2shows that GLO distribution is not a suitable candidate forthe L-moments and PL-moments
Interestingly both analyses of the L-moments and the PL-moments diagram show that the sample average values areappropriately distinguished by the hypothetical L-momentsand PL-moments for GPA and GNO distributions
However it is hard to find a suitable probability distri-bution that fits most of the regional observed data Table 5shows the goodness of fit test results for candidate probabilitydistributions
Table 5 shows that GLO distribution failed the goodnessof fit test for both L-moments and for PL-moments methodsas the calculated value of the 119885-test for the GLO distribution
Advances in Meteorology 9
Table 4 Heterogeneity measures for the study region based on L-moments and PL-moments
Observed standard deviation of group L-Cv 0078 0077Simulated mean of standard deviation of group L-Cv 0084 0077Simulated standard deviation of standard deviation of group L-Cv 0015 0014Value of the heterogeneity measure1198671 minus0410 minus0060
Heterogeneity measure1198672Observed average of L-CvL-skewness distance 0105 0139Simulated mean of average L-CvL-skewness distance 0141 0139Simulated standard deviation of average L-CvL-skewness distance 0023 0022Value of the heterogeneity measure1198672 minus1600 minus1800
Heterogeneity measure1198673Observed average of L-skewnessL-kurtosis distance 0094 0088Simulated mean of average L-skewnessL-kurtosis distance 0175 0176Simulated standard deviation of average L-skewnessL-kurtosis distance 0028 0028Value of the heterogeneity measure1198673 minus2890 minus3170
Table 5 119885-test result for the goodness of fitMethod GLO GEV GNO GPAL-moments 196 138 061 minus042PL-moments 177 119 043 minus061Table 6 Regional parameters for the three candidate distributionsfor L-moments and PL-moments
Method Distribution Parameters120585 120572 119870L-moments
is larger than the critical value of 164 (at 90 confidencelevel)
It has been observed that the computed values of |119885Dis|are less than 164 (at 90 confidence level) namely GEVGNO andGPA distributions However GEV GNO andGPAdistributions are suitable for regional distribution based onL-moments and PL-moments methods and for obtaining thefuture estimates of the quantile
Further it can be noted that GPA distribution is suitablefor L-moments method (lowest critical |119885Dis| value) Simi-larly GNO distribution is suitable for PL-moments method(lowest critical value) Table 6 shows the estimates of theregional parameters for L-moments and PL-moments for thesuitable probability distribution
33 Estimation of the Quantiles The regional quantile esti-mates 119902(119865) with varying nonexceedance probability119865 for theGNO GEV and GPA distributions are presented in Table 7
based on L-moments and PL-moments Quantile functionis normally represented as 119902(sdot) for fitted regional frequencydistribution The quantile estimate at location 119894 is establishedby joining the estimate of 120583119894 and 119902(sdot)
Mathematical form of the quantile estimate with nonex-ceedance probability 119865 is
The regional growth curves for the GEV GNO and GPAdistributions are shown in Figure 3
Figure 3 shows the regional growth curves of each can-didate distribution for L-moments and PL-moments GEVGNO and GPA distributions are approximately identicalup until 100-year return period (119865 = 099) for both L-moments and PL-moments However afterward the growthcurves of the GPA distribution lie below the GEV and GNOdistributions
Therefore it is necessary to assess the performance ofregional quantile estimates
4 Accuracy of the Estimated Quantiles and theRegional Growth Curve
A Monte Carlo simulation is designed to assess accuracyof the regional quantile estimates that are obtained by theRFA We use logical L-moments algorithm that has beenreported by Hosking and Wallis [7] in Section 64 Thisalgorithm takes samples from a region that has comparablecharacteristics as of the actual region such as having the samerecord length same number of locations and the regionalL-moments ratios The area used for simulation shouldreport the plausible heterogeneity in the area and intersitedependency if exist (Hosking andWallis [7]) In the repeatedsampling procedure the quantile estimates are computed forthe different nonexceedance probabilities Suppose that at119898th repetition and location 119894 quantile estimate can bewritten
10 Advances in Meteorology
Table 7 Regional quantile estimates with nonexceedance probability 119865Method Distribution 119865
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)
= 1119872 [[119872sum119898=1
(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12
(27)
Also for the estimated quantile the regional average bias andthe relative RMSE are
119861119877 (119865) = 1119873119873sum119894=1
119861119894 (119865)
119877119877 (119865) = 1119873119873sum119894=1
119877119894 (119865) (28)
We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval
119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)
Advances in Meteorology 11
Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le
119902 (119865)119871005 (119865) (30)
The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves
We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]
We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively
Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds
Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion
This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail
GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments
Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents
Appendix
The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows
PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by
119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
For the present study area realization of the Kappaprobability distribution is used to conduct the heterogeneitytest based on the L-moments and PL-moments
Number of simulations are 10000 for computing theheterogeneityWe computed the regional average L-momentsratios the regional PL-moments ratios and the correspond-ing parameter values of the fitted Kappa probability dis-tribution (see Table 4) Table 4 shows the results of theheterogeneity measure using L-moments and PL-momentsmethods It can be observed from Table 4 that the different
values for the119867-statistic are minus041 minus160 and minus289 based onL-moments andminus006minus18 andminus317 based onPL-momentsTherefore we concluded that by comparing these resultsand the heterogeneity conditions study region is acceptablyhomogeneous for L-moments and PL-moments No furthersubdivisions of the present study are necessary
32 Fitting Appropriate Probability Distribution After homo-geneity analysis of the study area a suitable probabilitydistribution is required for the RFAThe objective is not onlyto recognize a suitable probability distribution for RFA butalso to observe a probability distribution that will providerobust quantile estimate for each location and for the regionalgrowth cure List of candidate probability distributions forRFA is GLO GEV GPA and GNO
We plotted L-moments and PL-moments diagrams forpreliminary evaluation of the probability distribution for thestudy area
Figure 2 illustrates an analogy of the observed and hypo-thetical relationships of the probability distribution Figure 2shows that GLO distribution is not a suitable candidate forthe L-moments and PL-moments
Interestingly both analyses of the L-moments and the PL-moments diagram show that the sample average values areappropriately distinguished by the hypothetical L-momentsand PL-moments for GPA and GNO distributions
However it is hard to find a suitable probability distri-bution that fits most of the regional observed data Table 5shows the goodness of fit test results for candidate probabilitydistributions
Table 5 shows that GLO distribution failed the goodnessof fit test for both L-moments and for PL-moments methodsas the calculated value of the 119885-test for the GLO distribution
Advances in Meteorology 9
Table 4 Heterogeneity measures for the study region based on L-moments and PL-moments
Observed standard deviation of group L-Cv 0078 0077Simulated mean of standard deviation of group L-Cv 0084 0077Simulated standard deviation of standard deviation of group L-Cv 0015 0014Value of the heterogeneity measure1198671 minus0410 minus0060
Heterogeneity measure1198672Observed average of L-CvL-skewness distance 0105 0139Simulated mean of average L-CvL-skewness distance 0141 0139Simulated standard deviation of average L-CvL-skewness distance 0023 0022Value of the heterogeneity measure1198672 minus1600 minus1800
Heterogeneity measure1198673Observed average of L-skewnessL-kurtosis distance 0094 0088Simulated mean of average L-skewnessL-kurtosis distance 0175 0176Simulated standard deviation of average L-skewnessL-kurtosis distance 0028 0028Value of the heterogeneity measure1198673 minus2890 minus3170
Table 5 119885-test result for the goodness of fitMethod GLO GEV GNO GPAL-moments 196 138 061 minus042PL-moments 177 119 043 minus061Table 6 Regional parameters for the three candidate distributionsfor L-moments and PL-moments
Method Distribution Parameters120585 120572 119870L-moments
is larger than the critical value of 164 (at 90 confidencelevel)
It has been observed that the computed values of |119885Dis|are less than 164 (at 90 confidence level) namely GEVGNO andGPA distributions However GEV GNO andGPAdistributions are suitable for regional distribution based onL-moments and PL-moments methods and for obtaining thefuture estimates of the quantile
Further it can be noted that GPA distribution is suitablefor L-moments method (lowest critical |119885Dis| value) Simi-larly GNO distribution is suitable for PL-moments method(lowest critical value) Table 6 shows the estimates of theregional parameters for L-moments and PL-moments for thesuitable probability distribution
33 Estimation of the Quantiles The regional quantile esti-mates 119902(119865) with varying nonexceedance probability119865 for theGNO GEV and GPA distributions are presented in Table 7
based on L-moments and PL-moments Quantile functionis normally represented as 119902(sdot) for fitted regional frequencydistribution The quantile estimate at location 119894 is establishedby joining the estimate of 120583119894 and 119902(sdot)
Mathematical form of the quantile estimate with nonex-ceedance probability 119865 is
The regional growth curves for the GEV GNO and GPAdistributions are shown in Figure 3
Figure 3 shows the regional growth curves of each can-didate distribution for L-moments and PL-moments GEVGNO and GPA distributions are approximately identicalup until 100-year return period (119865 = 099) for both L-moments and PL-moments However afterward the growthcurves of the GPA distribution lie below the GEV and GNOdistributions
Therefore it is necessary to assess the performance ofregional quantile estimates
4 Accuracy of the Estimated Quantiles and theRegional Growth Curve
A Monte Carlo simulation is designed to assess accuracyof the regional quantile estimates that are obtained by theRFA We use logical L-moments algorithm that has beenreported by Hosking and Wallis [7] in Section 64 Thisalgorithm takes samples from a region that has comparablecharacteristics as of the actual region such as having the samerecord length same number of locations and the regionalL-moments ratios The area used for simulation shouldreport the plausible heterogeneity in the area and intersitedependency if exist (Hosking andWallis [7]) In the repeatedsampling procedure the quantile estimates are computed forthe different nonexceedance probabilities Suppose that at119898th repetition and location 119894 quantile estimate can bewritten
10 Advances in Meteorology
Table 7 Regional quantile estimates with nonexceedance probability 119865Method Distribution 119865
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)
= 1119872 [[119872sum119898=1
(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12
(27)
Also for the estimated quantile the regional average bias andthe relative RMSE are
119861119877 (119865) = 1119873119873sum119894=1
119861119894 (119865)
119877119877 (119865) = 1119873119873sum119894=1
119877119894 (119865) (28)
We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval
119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)
Advances in Meteorology 11
Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le
119902 (119865)119871005 (119865) (30)
The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves
We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]
We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively
Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds
Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion
This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail
GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments
Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents
Appendix
The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows
PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by
119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
For the present study area realization of the Kappaprobability distribution is used to conduct the heterogeneitytest based on the L-moments and PL-moments
Number of simulations are 10000 for computing theheterogeneityWe computed the regional average L-momentsratios the regional PL-moments ratios and the correspond-ing parameter values of the fitted Kappa probability dis-tribution (see Table 4) Table 4 shows the results of theheterogeneity measure using L-moments and PL-momentsmethods It can be observed from Table 4 that the different
values for the119867-statistic are minus041 minus160 and minus289 based onL-moments andminus006minus18 andminus317 based onPL-momentsTherefore we concluded that by comparing these resultsand the heterogeneity conditions study region is acceptablyhomogeneous for L-moments and PL-moments No furthersubdivisions of the present study are necessary
32 Fitting Appropriate Probability Distribution After homo-geneity analysis of the study area a suitable probabilitydistribution is required for the RFAThe objective is not onlyto recognize a suitable probability distribution for RFA butalso to observe a probability distribution that will providerobust quantile estimate for each location and for the regionalgrowth cure List of candidate probability distributions forRFA is GLO GEV GPA and GNO
We plotted L-moments and PL-moments diagrams forpreliminary evaluation of the probability distribution for thestudy area
Figure 2 illustrates an analogy of the observed and hypo-thetical relationships of the probability distribution Figure 2shows that GLO distribution is not a suitable candidate forthe L-moments and PL-moments
Interestingly both analyses of the L-moments and the PL-moments diagram show that the sample average values areappropriately distinguished by the hypothetical L-momentsand PL-moments for GPA and GNO distributions
However it is hard to find a suitable probability distri-bution that fits most of the regional observed data Table 5shows the goodness of fit test results for candidate probabilitydistributions
Table 5 shows that GLO distribution failed the goodnessof fit test for both L-moments and for PL-moments methodsas the calculated value of the 119885-test for the GLO distribution
Advances in Meteorology 9
Table 4 Heterogeneity measures for the study region based on L-moments and PL-moments
Observed standard deviation of group L-Cv 0078 0077Simulated mean of standard deviation of group L-Cv 0084 0077Simulated standard deviation of standard deviation of group L-Cv 0015 0014Value of the heterogeneity measure1198671 minus0410 minus0060
Heterogeneity measure1198672Observed average of L-CvL-skewness distance 0105 0139Simulated mean of average L-CvL-skewness distance 0141 0139Simulated standard deviation of average L-CvL-skewness distance 0023 0022Value of the heterogeneity measure1198672 minus1600 minus1800
Heterogeneity measure1198673Observed average of L-skewnessL-kurtosis distance 0094 0088Simulated mean of average L-skewnessL-kurtosis distance 0175 0176Simulated standard deviation of average L-skewnessL-kurtosis distance 0028 0028Value of the heterogeneity measure1198673 minus2890 minus3170
Table 5 119885-test result for the goodness of fitMethod GLO GEV GNO GPAL-moments 196 138 061 minus042PL-moments 177 119 043 minus061Table 6 Regional parameters for the three candidate distributionsfor L-moments and PL-moments
Method Distribution Parameters120585 120572 119870L-moments
is larger than the critical value of 164 (at 90 confidencelevel)
It has been observed that the computed values of |119885Dis|are less than 164 (at 90 confidence level) namely GEVGNO andGPA distributions However GEV GNO andGPAdistributions are suitable for regional distribution based onL-moments and PL-moments methods and for obtaining thefuture estimates of the quantile
Further it can be noted that GPA distribution is suitablefor L-moments method (lowest critical |119885Dis| value) Simi-larly GNO distribution is suitable for PL-moments method(lowest critical value) Table 6 shows the estimates of theregional parameters for L-moments and PL-moments for thesuitable probability distribution
33 Estimation of the Quantiles The regional quantile esti-mates 119902(119865) with varying nonexceedance probability119865 for theGNO GEV and GPA distributions are presented in Table 7
based on L-moments and PL-moments Quantile functionis normally represented as 119902(sdot) for fitted regional frequencydistribution The quantile estimate at location 119894 is establishedby joining the estimate of 120583119894 and 119902(sdot)
Mathematical form of the quantile estimate with nonex-ceedance probability 119865 is
The regional growth curves for the GEV GNO and GPAdistributions are shown in Figure 3
Figure 3 shows the regional growth curves of each can-didate distribution for L-moments and PL-moments GEVGNO and GPA distributions are approximately identicalup until 100-year return period (119865 = 099) for both L-moments and PL-moments However afterward the growthcurves of the GPA distribution lie below the GEV and GNOdistributions
Therefore it is necessary to assess the performance ofregional quantile estimates
4 Accuracy of the Estimated Quantiles and theRegional Growth Curve
A Monte Carlo simulation is designed to assess accuracyof the regional quantile estimates that are obtained by theRFA We use logical L-moments algorithm that has beenreported by Hosking and Wallis [7] in Section 64 Thisalgorithm takes samples from a region that has comparablecharacteristics as of the actual region such as having the samerecord length same number of locations and the regionalL-moments ratios The area used for simulation shouldreport the plausible heterogeneity in the area and intersitedependency if exist (Hosking andWallis [7]) In the repeatedsampling procedure the quantile estimates are computed forthe different nonexceedance probabilities Suppose that at119898th repetition and location 119894 quantile estimate can bewritten
10 Advances in Meteorology
Table 7 Regional quantile estimates with nonexceedance probability 119865Method Distribution 119865
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)
= 1119872 [[119872sum119898=1
(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12
(27)
Also for the estimated quantile the regional average bias andthe relative RMSE are
119861119877 (119865) = 1119873119873sum119894=1
119861119894 (119865)
119877119877 (119865) = 1119873119873sum119894=1
119877119894 (119865) (28)
We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval
119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)
Advances in Meteorology 11
Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le
119902 (119865)119871005 (119865) (30)
The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves
We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]
We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively
Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds
Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion
This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail
GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments
Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents
Appendix
The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows
PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by
119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
Observed standard deviation of group L-Cv 0078 0077Simulated mean of standard deviation of group L-Cv 0084 0077Simulated standard deviation of standard deviation of group L-Cv 0015 0014Value of the heterogeneity measure1198671 minus0410 minus0060
Heterogeneity measure1198672Observed average of L-CvL-skewness distance 0105 0139Simulated mean of average L-CvL-skewness distance 0141 0139Simulated standard deviation of average L-CvL-skewness distance 0023 0022Value of the heterogeneity measure1198672 minus1600 minus1800
Heterogeneity measure1198673Observed average of L-skewnessL-kurtosis distance 0094 0088Simulated mean of average L-skewnessL-kurtosis distance 0175 0176Simulated standard deviation of average L-skewnessL-kurtosis distance 0028 0028Value of the heterogeneity measure1198673 minus2890 minus3170
Table 5 119885-test result for the goodness of fitMethod GLO GEV GNO GPAL-moments 196 138 061 minus042PL-moments 177 119 043 minus061Table 6 Regional parameters for the three candidate distributionsfor L-moments and PL-moments
Method Distribution Parameters120585 120572 119870L-moments
is larger than the critical value of 164 (at 90 confidencelevel)
It has been observed that the computed values of |119885Dis|are less than 164 (at 90 confidence level) namely GEVGNO andGPA distributions However GEV GNO andGPAdistributions are suitable for regional distribution based onL-moments and PL-moments methods and for obtaining thefuture estimates of the quantile
Further it can be noted that GPA distribution is suitablefor L-moments method (lowest critical |119885Dis| value) Simi-larly GNO distribution is suitable for PL-moments method(lowest critical value) Table 6 shows the estimates of theregional parameters for L-moments and PL-moments for thesuitable probability distribution
33 Estimation of the Quantiles The regional quantile esti-mates 119902(119865) with varying nonexceedance probability119865 for theGNO GEV and GPA distributions are presented in Table 7
based on L-moments and PL-moments Quantile functionis normally represented as 119902(sdot) for fitted regional frequencydistribution The quantile estimate at location 119894 is establishedby joining the estimate of 120583119894 and 119902(sdot)
Mathematical form of the quantile estimate with nonex-ceedance probability 119865 is
The regional growth curves for the GEV GNO and GPAdistributions are shown in Figure 3
Figure 3 shows the regional growth curves of each can-didate distribution for L-moments and PL-moments GEVGNO and GPA distributions are approximately identicalup until 100-year return period (119865 = 099) for both L-moments and PL-moments However afterward the growthcurves of the GPA distribution lie below the GEV and GNOdistributions
Therefore it is necessary to assess the performance ofregional quantile estimates
4 Accuracy of the Estimated Quantiles and theRegional Growth Curve
A Monte Carlo simulation is designed to assess accuracyof the regional quantile estimates that are obtained by theRFA We use logical L-moments algorithm that has beenreported by Hosking and Wallis [7] in Section 64 Thisalgorithm takes samples from a region that has comparablecharacteristics as of the actual region such as having the samerecord length same number of locations and the regionalL-moments ratios The area used for simulation shouldreport the plausible heterogeneity in the area and intersitedependency if exist (Hosking andWallis [7]) In the repeatedsampling procedure the quantile estimates are computed forthe different nonexceedance probabilities Suppose that at119898th repetition and location 119894 quantile estimate can bewritten
10 Advances in Meteorology
Table 7 Regional quantile estimates with nonexceedance probability 119865Method Distribution 119865
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)
= 1119872 [[119872sum119898=1
(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12
(27)
Also for the estimated quantile the regional average bias andthe relative RMSE are
119861119877 (119865) = 1119873119873sum119894=1
119861119894 (119865)
119877119877 (119865) = 1119873119873sum119894=1
119877119894 (119865) (28)
We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval
119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)
Advances in Meteorology 11
Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le
119902 (119865)119871005 (119865) (30)
The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves
We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]
We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively
Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds
Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion
This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail
GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments
Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents
Appendix
The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows
PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by
119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)
= 1119872 [[119872sum119898=1
(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12
(27)
Also for the estimated quantile the regional average bias andthe relative RMSE are
119861119877 (119865) = 1119873119873sum119894=1
119861119894 (119865)
119877119877 (119865) = 1119873119873sum119894=1
119877119894 (119865) (28)
We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval
119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)
Advances in Meteorology 11
Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le
119902 (119865)119871005 (119865) (30)
The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves
We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]
We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively
Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds
Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion
This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail
GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments
Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents
Appendix
The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows
PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by
119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le
119902 (119865)119871005 (119865) (30)
The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves
We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]
We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively
Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds
Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion
This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail
GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments
Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents
Appendix
The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows
PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by
119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012