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Hydrological Sciences-Journal-des Sciences Hydrologiques,40,2,April 1995 1 6 5
Parameter estimation for 3-parameter
generalized pareto distribution by the
principle of maximum entropy (POME)
V. P. SINGH & H. GUO
DepartmentofCivilEngineering,Louisiana State University, BatonRouge,
Louisiana 70803-6405, USA
Abstract
The principle of maximum entropy (POME) is employed to
derive a new method of parameter estimation for the 3-parameter
generalized Pareto (GP) distribution. Monte Carlo simulated data are
used to evaluate this method and compare it with the methods of
moments (MOM), probability weighted moments (PW M), and maximum
likelihood estimation (MLE). The parameter estimates yielded by the
POME are either superior or comparable for high skewness.
Estimation des paramtres d'une loi de Pareto gnralise
trois paramtres par la mthode du maximum d'entropie
Rsum
Nous avons utilis le principe du maximum d'entropie en vue
d'tablir une nouvelle mthode d'estimation des paramtres de la
distribution de Pareto gnralise trois paramtres. Des donnes
synthtiques gnres selon une procdure de Monte Carlo ont t
utilises pour valuer cette mthode et pour la comparer aux mthodes
des moments, des moments pondrs et du maximum de vraisemblance.
L'estimation des paramtres s'appuyant sur le principe du maximum
d'entropie est prfrable ou comparable celle des autres mthodes en
particulier lorsque l'asymtrie est forte.
G EN ERA LIZED P A RETO D IS TRIBU TIO N
Consider a random variable Y with the standard exponential distribution. Let
a random variable Xbe defined as X = b{\
exp(-aY))/a, whereaandb are
parameters. Then the distribution of X is the 2-parameter generalized Pareto
distribution. If c is the threshold or lower bound ofX, then the distribution of
X
is the 3-parameter generalized Pareto (GP) distribution which can be
expressed as:
F(x)
=
1
-
1 -
=
1
- exp
a(x
b
x
c)
c
a jt
0
a = 0
(la)
( lb)
where cis a location parameter, b is a scale parameter, a is a shape parameter,
Openfordiscussion until 1 October 1995
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166
V. P. Singh N. Guo
andF{x)is the distribution function. The probability density function (PDF) of
the GP distribution is given by:
m
1
b
1 -
exp
aixc)
b
xc
b
a * 0
0
(2a)
(2b)
The Pareto distributions are obtained for a < 0. Figure1shows the PD F
for c = 0, b = 1.0, and various values of a. P ickands (1975) has shown that
the GP distribution given by equation (1) occurs as a limiting distribution for
excesses over thresholds if and only if the parent distribution is in the domain
of attraction of one of the extreme value distributions. The GP distribution
reduces to the 2-parameter GP distribution for c = 0, the exponential distri
bution for a 0 and c = 0, and the uniform distribution on [0,b]for c = 0
and(3= 1.
b)
z
o
rj
>
0.5-
< 7
-
4
o
0.3
0.0 0 .2 0.4 0.6 08 1.0 1.2 1.4 1.6
Line: a = 0.5; plus: a = 0.7 5;
star: a = 1.0; and dash: a = 1.25
0.0 0.2 0.4 0.6 0.8
1.2 1.4 1.6 1.8 2.0
Line: a = - 0 . 1 ; d a s h : a = - 0 . 5 ;
p lus : a = - 1 . 0
Fig. 1 Probability density function of generalized Pareto distribution with
(a)c = 0, b = 1.0, a = 0.5, 0.75, 1.0 and 1.25; and (b) with c = 0, b =
1.0,
a
= - 0 . 1 , - 0 . 5 and - 1 . 0 .
Some important properties of the GP distribution are worth mentioning:
(1) By com parison with the expon ential distribution, the GP distribution has
a heavier tail for a 0 (short-tailed distribution). When a
0; and
c
c,corresponding to a higher threshold Q
0
+ calso has a GP distri
bution. This is one of the properties that justifies the use of GP
distribution to model excesses.
Let Z = max(c, X
{
, X
2
, ..., X
N
), whereN > 0 is a number. IfX
h
i =
1,
2, .. . , N, are independent and identically distributed as a GP
distribution, and
N
has a Poisson distribution, then Z has a generalized
extreme value distribution (GEV) (Smith, 1984; Jin & Stedinger, 1989;
Wang, 1990), as defined by Jenkinson (1955). Thus, a Poisson process
of exceedance times with generalized Pareto excesses implies the
classical extreme value distributions. As a special case, the maximum of
a Poisson number of exponential varites lias a Gumbel distribution. So
exponential peaks lead to Gumbel maxima, and GP distribution peaks
lead to GEV maxima. The GEV can be expressed as:
F Z)
exp
exp
\l-
Z
~
y
13
-
exp
z-y
(3
l
I
-
-, -
0,
z
> 0
0
(3a)
(3b)
where the parameters
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V.
P.
Singh &N. Guo
reliability studies and the analysis of environmental extremes. Davison Smith
(1990) pointed out that the GP distribution might form the basis of a broad
modelling approach to high-level exceedances. DuMouchel (1983) applied it to
estimate the stable indexato measure tail thickness, whereas Davison (1984a,
1984b) modelled contamination due to long-range atmospheric transport of
radionuclides, van Montfort & Witter (1985, 1986) and van Montfort & Otten
(1991) applied the GP distribution to model the peaks over a threshold (POT)
streamflows and rainfall series, and Smith (1984, 1987, 1991) applied it to
analyse flood frequencies and wave heights. Similarly, Joe (1987) employed it
to estimate quantiles of the maximum of
iVobservations.
Wang (1991) applied
it to develop a POT model for flood peaks with Poisson arrival time, whereas
Rosbjerget
al.
(1992) compared the use ofthe2-parameter GP and exponential
distributions as distributionmodelsfor exceedances with the parent distribution
being a generalized GP distribution. In an extreme value analysis of the flow
of Burbage Brook, Barrett (1992) used the GP distribution to model the POT
flood series with Poisson inter-arrival times. Davison Smith (1990) presented
a comprehensive analysis of the extremes of data by use of the GP distribution
for modelling the sizes and occurrences of exceedances over high thresholds.
Methods for estimatingtheparameters of the 2-parameter GP distribution
were reviewed by Hosking & Wallis (1987). Quandt (1966) used the method
of moments (MOM ), while Baxter (1980) and Cook Mumme (1981) used the
method of maximum likelihood estimation (MLE) for the Pareto distribution.
The MOM, MLE and probability weighted moments (PWM) were included in
the review, van Montfort & Witter (1986) used the MLE to fit the GP distri
bution to represent the Dutch POT rainfall series and used an empirical
correction formula to reduce bias of the scale and shape parameter estimates.
Davison & Smith (1990) used the MLE, PWM, a graphical method and least
squares to estimate the GP distribution parameters. Wang (1991) derived the
PWM for both known and unknown thresholds.
OBJECTIVE OF
STUDY
The objective of this paper is to develop a new competitive method of
parameter estimation based on the principle of maximum entropy (POME), and
to compare it with the MOM, MLE and PWM using Monte Carlo simulated
data. The review of the literature shows that the POME does not appear to
have been employed for estimating parameters of the GP distribution.
DERIVATION OF PARAMETER ESTIMATION METHOD BY
POME
Shannon (1948) defined entropy as a numerical measure of uncertainty, or
conversely the information content associated with a probability distribution,
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Parameter estimation
for
generalized Pareto distribution 169
f(x;8), with a parameter vector0and usedtodescribearandom variableX.The
Shannon entropy function
H(f)
for continuousX canbeexpressedas:
H(f)
= -
fl.x;6) \nf(x;0)x
with
[/(x;0)dx
= l
4)
whereH f)is theentropy
off(x;0),
and can bethoughtof as themean valueof
-\nf(x;d).
AccordingtoJaynes (1961),theminimally biased distributionofXisthe
one which maximizes entropy subjectto given information, or which satisfies
the principleof maximum entropy (POME). Therefore, theparameters of the
distribution canbeobtainedbyachievingthem aximumof
H(f).
The useofthis
principlefor generatingthe least-biased probability distributionson thebasis
of limited
and
incomplete data
has
been discussed
by
several authors
and has
been applied to many diverse problems (e.g. a recent review by Singh
Fiorentino (1992)). Jaynes (1968)has reasoned thatthePOMEis thelogical
and rational criterionfor choosing some specific
f(x;d)
that maximizes Hand
satisfies the given information expressed as constraints. In other words, for
given information (e.g. mean, variance, skewness, lower limit, upper limit,
etc.),
thedistribution derivedby thePOM E w ould best represent
X;
implicitly,
this distribution would best represent the sample from whichthe information
was derived. Inversely,if it isdesiredto fit aparticular probab ility distribution
to
a
sample
of
data, then the POME can uniquely specify
the
constraints
(or the
information) neededtoderive that distribution. T he distribution parametersare
then related to these constraints. An excellent discussion of the underlying
mathematical rationaleisgiveninLevine Tribus (1979).
Givenmlinearly independent constraintsC
h
i = 1,2, ...,m,inthe form
C. =
\wfx)f{x;6)x,
i = 1,2,...,m (5)
where
w
t
(x )
are some functions whose averages
over f(x;6)
arespecified, then
the maximumofH subjecttoequation(5) isgivenby thedistribution:
f(x;6) = exp
-a
0
~ a,-w,-(x)
(=i
(6a)
where a
h
i = 0, 1, 2, ..., m, are the Lagrange multipliers, and can be
determined from equations(5) and (6a). Inserting equation (6a)inequation(4)
yieldstheentropy
of (x;6)
intermsofthe constraintsandLagrange mu ltipliers:
m
H(f) =
+
Y
j
a,C
i
(6b)
MaximizationofHthen establishestherelationships betw een constraints
and Lagrange multipliers. Thus,toderive a method usingthePOMEfor the
estimation of the parameters a, b and c of equation (2), three steps are
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170
V.
P. Singh N. Guo
involved: (i) specification of the appropriate constraints; (ii) derivation of the
entropy of the distribution; and (iii) derivation of the relationships between the
Lagrange multipliers and constraints. A complete mathematical discussion of
this method can be found in Tribus (1969), Jaynes (1968), Levine & Tribus
(1979) and Singh & Rajagopal (1986).
Specification of constraints
The entropy of the GP distribution can be derived by inserting equation (1) in
equation (4):
H(f) = lnof/fr ;0)dc-
1-1
a
In
. __a(x
c)
_
f(x;d)dx (
6 c
)
Comparing equation (6c) with equation (6b), the constraints appropriate for
equation (3) can be written (Singh & Rajagopal, 1986) as:
\f{x;d)
x = 1
7)
In
, _
a(xc)
f(x;6)dx = E In
1
_ a(xc)
b
(8)
in which E[*] denotes expectation of the bracketed quantity. These constraints
are unique and specify the information that is sufficient for the GP distribution.
The first constraint specifies the total probability. The second constraint
specifies the mean of the logarithm of the inverse ratio of the scale parameter
to the failure rate. Conceptually, this defines the expected value of the negative
logarithm of the scaled failure rate. The distribution parameters are related to
these constraints.
Construction
of
th e
entrop y function
The PDF of the GP distribution corresponding to the POME and consistent
with equations (7) and (8) takes the form:
f{x;d) =exp
a
Q
fljln
1
a(x - c)
(9)
where a
Q
and a
x
are Lagrange multipliers. The mathematical rationale for
equation (9) has been presented by Tribus (1969).
By applying equation (3) to the total probability condition in equation
(7),
one obtains:
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exp(a
0
)
Parameter estimationforgeneralized Pareto distribution
a(x - c)
exp
-jln
1 - . dx
which yields the partition function:
exp(a
0
) = -
1
a
1
- a ,
The zeroth Lagrange multiplier is given by:
a
0
= In
b 1
a Ia,
Inserting equation (11) in equation (9) yields:
Ax-B)
a(\ a,)
1 -
a(xc)
A comparison of equation (13) with equation (3) yields:
1
Ia, =
a
Taking logarithms of equation (13) gives:
lnf(x;d) = l na+ l n ( l - a
x
) -Inb-a^n
aix - c)
b
Therefore, the entropyH(J) of the GP distribution follows:
H(f) = lna ln(l a{) +lnb+a
l
E\ In
1 -
a(xc)
111
(10)
(11)
(12)
(13)
(14)
(15)
(16)
Relationships between distr ibution parameters and constraints
According to Singh & Rajagopal (1986), the relationships between the
distribution parameters and constraints are obtained by taking partial derivatives
of the entropy
H(f)
with respect to the Lagrange multipliers as well as the
distribution parameters, and then equating these derivatives to ze ro, and m aking
use of the constraints. To that end, taking partial derivatives of equation (16)
with respect to a
x
, a, b and c separately and equating each derivative to zero
yields:
dH
da,
1
I
a,
E
In 1 a(x - c)
_
0
dH
da
=
-~a
t
E
(x - c)lb
1 - a(x - c)lb
0
(17)
(18)
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V. P. Singh N. Guo
=
2
~db 1
dH
dc
= a j E
j j - E
(x - c)/6
1a(x
-
1
1a(xc)lb
-c)lb
= 0
(19)
(20)
Simplification of equations (17) to (20) yields, respectively:
1
In
1
a(xc)
b
(x - c)lb
1 a(xc)/b
(x
c)lb
1
a(x
c)/ft
1
I-a,
aa,
aa,
1 - a(xc)lb
(21)
22)
23)
24)
Clearly, equation (24) does not hold. Equation (22) is the same as equation
(23).
In order to get a unique solution, additional equations are needed which
can be obtained by differentiating the zeroth Lagrange multiplier with respect
to the Lagrange multipliers and equating the derivatives to zero. To that end,
equation (10) is written as:
a
Q
=
In exp fljln
1 -
a(xc)
dx
(25)
Differentiating equation (25) with respect to a
x
:
00
exp{fljln[l
a(xc)/b]}ln[l a(xc)/b]dx
da,
exp[a
0
ln{l a(xc)/b}]dx
^{-o.-aMl-^-cVbmi-aix-Omx
-E{[1-a(x-c)/b]}
(26)
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Parameter estimation
for
generalized Pareto distribution
Following Tribus (1969):
var{ln[l
- a(x- c)/b]}
da,
173
27)
where var[] is the variance of the bracketed quantity. From equation (11):
a
0
=\n b/a)-\n l-a
l
) (28)
Differentiating equation (28) with respect to
a
{
:
(29)
30)
da
0
_
da
x
d \
1
1 flj
1
da,
a-^r
Equating equation (29) to equation (26) leads to:
In 1 -
a(x- c)
b
1
I
a,
(31)
which is the same as equation (21). When equation (30) is equated to equation
(27),
the following is obtained:
var
In
1
a(x
c)
1
( l - ^ )
2
32)
Therefore, theparam eter estimation equations for the POM E consistof
equations (21), (22) and (32). Inserting a
x
= 1 - lia from equation (14) into
these three equations, one gets:
1 -
a(x
c)
1
1
a(xc)lb
var
In
a(x
c)
_
=
a
I a
= a
33)
34)
35)
T H R E E O T H E R M E T H O D S O F P A R A M E T E R E S T I M A T I O N
Three of the most popular methods of parameter estimation are the method of
moments (MOM), the methodofprobability-weighted m oments (PW M ), and
the method ofmaximum likelihood estimation (M LE ). Th e POM E does not
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V, P. Singh N. Guo
appear to have been used for estimating parameters of the GP distribution.
Therefore, virtually no literature exists on the comparison of parameter
estimates by the POME with those by the MLE, PWM and MOM. For the
sake of completeness, these methods are briefly summarized.
Method of moments (MOM)
Moment estimators of the GP distribution were derived by Hosking & Wallis
(1987). No te that E (l -
a(x
-
c)lb)
r
=
1/(1 +
ar)
if 1 + ra > 0. The rth
moment of X exists if a > - 1 / r . Provided that they exist, then the moment
estimators are:
x = c+Ji-
(36)
l+a
9
b
2
S
2
=
(37)
( l + f l )
2
( l +2a )
G = 2 ( l - Q ) ( l + 2 f l )
0
-
5
( 3 8 )
1
+3a
where x, S
2
and G are the mean, variance and skewness, respectively. First,
the moment estimate of a is obtained by solving equation (38). The relation
between G and
a
is illustrated in Fig. 2. With
a
calculated,
b
and
c
follow
from equation (36) and (37) as:
b =
S l+a) l+2af
5
(39)
c = x-- (40)
b+a
Probability-weighted moments
(PWM)
The PWM estimators for the GP distribution (Hosking & Wallis, 1987) are
given as:
a -
W
o~
SW
i-
9W
2
(41)
- W
0
+ 4W
1
3W
2
b = (
W
o-
2W
J(
W
o-
3W
2)(-4W
l+
6W
2
)
( - W
0
+ 4 W j - 3 F
2
)
2
2 W
Q
F
1
-6 W
0
W
2 +
6W
1
W
2
~W
0
+4W
l
-3W
2
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Para meter estimation for generalized Pareto distribution
175
i.o-
0.8
1
xi 0.6
CC
0.4-
t n o . 2 i
< 0-0
a:
< -0.2
-0.4-1
-0.6
-0.8
-1.0
o i :
SKEWNESS G
Fig. 2 Parametera vsskewness Gfor GPD3.
where the rth probability-weighted moment
W
r
is:
l
W
r
= E[x(F)(l~F(x)Y] =
{c
+
- [ l - ( l - F )
a
] } ( l - f ) ' ' d F
1
r + 1
ft
1
a a+r+1
r = 0 ,1 ,2 , . .
44)
Method of maximum likelihood estimation
The MLE estimators can be expressed as:
j , Xj-cVb
=
^ _
frf
1
a(x
(
. -
c)lb Ia
45)
J2 ln[l - a(x
(
- c)/b] = na
(46)
A maximum likelihood estimator cannot be obtained for c, because the
likelihood function is unbounded with respect to c, as shown in Fig. 3. Since
c is the lower bound of the random variable X, we may use the constraint
c
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V. P. Singh N. Guo
g
o
ZD
CL
8
X
_ l
UJ
o-
- i -
-2-
-3 -
-4 -
-5-
-6 -
-7-
- 8 -
- 9 -
-10-
- I I -
-12-
-13-
-14-
-15-^
OJO
0.1 0.2 0 3 0 .4 0.5
PARAMETER c
0.6
0.7
L in e: a = - 0 .1 1 6 , b 0.387 , c = 0 .562;
dash:
a 0 .544 , b = 1.116 c = 0.277
Fig . 3 Likelihood function of GPD 3 vs parameter c for sample size 10.
APPLICATION TO MONTE CARLO-SIMULATED DATA
Monte Carlo samples
To assess the performance ofthe POME estimation method by comparison with
the MOM, PWM and MLE, Monte Carlo sampling experiments were con
ducted. Two distribution population cases, listed in Table 1, were considered.
For each population case, 1000 random samples of size 20, 50 and 100 were
generated, and then parameters and quantiles were estimated.
Table 1 GP distribution population cases considered in the sampling
exper iment
GP distribution
population
Case 1
Case 2
c
v
0.5
0.5
G
0.5
2.5
Parameters
a b
0.554 1.116
- 0 .069 0 .433
c
0.277
0.536
C = coefficient of variation.
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Parameter estimationforgeneralized Pareto distribution 111
Performance indices
The performance of the POM E was evaluated using the following performance
indices:
Standard bias BIAS =
(*>~* (47)
x
Root mean square error RMSE =
E
K
X - X
)
1
' (48)
A
, 2 i0 .5
X
wherex is an estimate ofx (parameter or quantile) and:
N
W) = jt*i
N
i=
i
whereTVis the number of Monte Carlo samples(N = 1000 in this study). 1000
may arguably not be a large enough number of samples to produce the true
values of BIAS and RMSE, but will suffice to compare the performances of the
estimation methods.
BIAS in parameter estimation
The bias of parameters estimated by the four methods is summarized in
Table 2. For G = 0.5, in absolute terms the MOM produced the least bias of
the four methods for all sample sizes. The MLE had the second least bias in
the parameter estimates. With increasing sample size, there was significant
reduction in bias for all four methods. The POME produced less bias than the
PWM in estimates ofb and c for all sample sizes, but that was not uniformly
true in the case of the estimate of parameter
a.
When G = 2.5, these methods
performed quite differently. For all samples sizes, the MLE and the POME
Table 2 BIAS of parameter estimates
Sample size
20
50
100
Method
MOM
PWM
MLE
POME
MOM
PWM
MLE
POME
M O M
PWM
MLE
POME
G =
0.5
a
0.156
0.488
0.217
- 0 . 3 9 7
0.063
0.230
0.132
- 0 . 4 0 7
0.040
0.132
0.086
- 0 . 2 8 8
b
0.094
0.632
0.037
- 0 . 1 2 2
0.042
0.258
0.060
- 0 . 0 9 6
0.028
0.138
0.048
- 0 . 0 6 0
c
- 0 . 0 5 3
- 0 . 9 4 8
0.215
- 0 . 0 9 4
- 0 . 0 2 5
- 0 . 3 9 6
0.067
- 0 . 1 5 6
- 0 . 0 1 9
- 0 . 2 0 8
0.039
- 0 . 1 2 6
G = 2.5
a
- 4 . 1 4 4
- 9 . 1 4 1
0.474
0.013
- 1 . 9 8 1
- 3 . 8 2 1
0.244
0.009
- 1 . 1 9 6
- 1 . 9 6 4
0.185
0.012
b
0.509
1.799
- 0 . 0 7 7
0.147
0.260
0.626
- 0 . 0 2 4
0.115
0.165
0.304
- 0 . 0 1 7
0.099
c
- 0 . 1 4 3
- 0 . 5 8 4
0.034
- 0 . 0 9 4
- 0 . 0 8 5
- 0 . 2 3 1
0.009
- 0 . 0 7 9
- 0 . 0 5 7
- 0 . 1 1 6
0.008
- 0 . 0 6 8
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V. P. Singh N. Guo
were comparable, producing the least bias. For the a and c parameter
estimates, the POME had the least bias, but the MLE had the least bias for the
b
parameter estimate. The PWM had the highest bias in all three parameter
estimates for all sample sizes. Thus, if the value of G is high, the POME or
MLE may be the preferred method. For lower values of G, the MOM or MLE
may be preferable, especially when the sample size is small.
RMSE in parameter estimation
The values of RMSE of parameters estimated by the four methods are given in
Table 3. For G = 0.5, of the four methods the MOM produced the least
RMSE in thea parameter estimate. However, as the sample size increased, the
MOM, PWM and MLE became comparable. In the cases of the b and cpara
meter estimates, the MLE had the least RMSE, but all four methods were
comparable. For G = 2.5, the comparative behaviour of the four methods was
markedly different. In absolute terms, the MOM and the PWM produced the
highest RMSE in parameter estimates for all sample sizes, with the POME
having the least bias in the
a
parameter estimate but the MLE in the
b
and
c
parameter estimates. Thus, it may be concluded that for lower values of G, the
MOM or PWM may be the preferred method, but for higher values of G, the
MLE or POME is the preferred method.
BIAS in quantile estimation
The results of bias in quantile estimates by the GP distribution are summarized
in Table 4. The performance of the four estimation methods varied with the
value of G, and probability of non-exceedance P. For G = 0.5, all four
methods had comparable bias for P
0.99, the MOM and the PWM produced the smallest bias and the POME the
Tab le 3 RMSE of parameter estimates
Method
M OM
PWM
MLE
POME
M O M
PWM
MLE
POME
MOM
PWM
MLE
POME
a
0.448
0.780
0.502
0.785
0.301
0.419
0.329
0.696
0.203
0.268
0.224
0.590
b
0.310
0.820
0.284
0.371
0.213
0.365
0.234
0.271
0.144
0.211
0.176
0.233
c
0.336
0.984
0.357
0.348
0.201
0.427
0.146
0.262
0.139
0.237
0.056
0.185
a
- 5 . 1 7 8
--10.990
- 1 . 9 2 6
- 0 . 0 6 7
- 2 . 7 8 5
- 4 . 8 3 0
- 1 . 4 7 5
- 0 . 0 6 1
- 1 . 9 2 5
- 2 . 7 1 0
- 1 . 2 0 5
- 0 . 0 6 1
b
0.688
2.005
2.580
0.394
0.376
0.710
0.177
0.250
0.249
0.360
0.127
0.181
c
0.205
0.593
0.053
0.182
0.120
0.236
0.019
0.125
0.083
0.121
0.011
0.097
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Param eter estimation for generalized Pa reto distribution 179
Tab le 4 BIAS and RMSE of quantile estimates
p
0.8
0.9
0.99
0.999
Sample size
20
50
100
20
50
100
20
50
100
20
50
100
Method
MOM
PWM
MLE
POME
MOM
PWM
MLE
POME
MOM
PWM
MLE
POME
MOM
PWM
MLE
POME
MOM
PWM
MLE
POME
MOM
PWM
MLE
POME
MOM
PWM
MLE
POME
MOM
PWM
MLE
POME
MOM
PWM
MLE
POME
MOM
PWM
MLE
POME
MOM
PWM
MLE
POME
MOM
PWM
MLE
POME
G
= 0.5
BIAS
0.000
0.091
- 0 . 0 1 1
- 0 . 0 3 0
0.001
0.041
0.010
0.000
- 0 . 0 1 2
0.076
- 0 . 0 3 7
0.031
- 0 . 0 2 1
0.153
- 0 . 0 8 2
- 0 . 0 6 2
- 0 . 0 0 4
0.032
- 0 . 0 0 5
0.066
0.000
0.018
0.004
0.048
- 0 . 0 2 6
0.036
- 0 . 0 6 7
0.287
- 0 . 0 0 9
0.007
- 0 . 0 2 9
0.323
- 0 . 0 0 5
0.002
0.014
0.230
- 0 . 0 2 2
0.028
- 0 . 0 6 3
0.582
- 0 . 0 0 5
0.000
- 0 . 0 3 4
0.612
- 0 . 0 0 4
- 0 . 0 0 4
- 0 . 0 1 9
0.439
RMSE
0.112
0.152
0.118
0.128
0.078
0.090
0.093
0.076
0.098
0.131
0.153
0.151
0.149
0.231
0.157
0.158
0.065
0.074
0.072
0.126
0.043
0.048
0.055
0.092
0.113
0.153
0.128
0.484
0.070
0.087
0.063
0.491
0.048
0.060
0.039
0.399
0.141
0.192
0.174
0.888
0.090
0.113
0.079
0.906
0.063
0.078
0.047
0.474
G =
2.5
BIAS
0.058
0.169
- 0 . 0 1 8
0.046
0.037
0.083
- 0 . 0 0 4
0.033
0.024
0.115
- 0 . 0 6 8
0.068
0.015
0.186
- 0 . 0 4 7
0.064
0.024
0.063
- 0 . 0 0 1
0.051
0.019
0.038
0.001
0.044
- 0 . 1 3 1
- 0 . 1 2 9
0.031
0.104
- 0 . 0 5 9
- 0 . 0 7 4
0.031
0.080
- 0 . 0 3 1
- 0 . 0 6 5
0.023
0.070
- 0 . 2 6 6
- 0 . 2 9 6
0.152
0.121
- 0 . 1 4 1
- 0 . 1 9 8
0.100
0.093
- 0 . 0 8 3
- 0 . 1 2 0
0.069
0.081
RMSE
0.172
0.224
0.134
0.176
0.107
0.125
0.090
0.109
0.197
0.225
0.221
0.221
0.273
0.348
0.224
0.271
0.123
0.131
0.106
0.137
0.084
0.085
0.073
0.098
0.309
0.372
0.286
0.297
0.205
0.235
0.203
0.186
0.154
0.165
0.150
0.135
0.427
0.600
0.572
0.332
0.310
0.393
0.406
0.207
0.252
0.289
0.295
0.151
highest, with the MLE in the intermediate range. However, for G = 2.5, the
POME produced the least bias, especially when P was greater than 0.99. For
all sample sizes, all four methods were somewhat comparable. In conclusion,
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180
V. P. Singh N. Guo
for lower values of G, anyone of the four methods may be used for P 0.99 , the performance of the
POME deteriorated. When
G =
2.5, all methods produced comparable values
of RMSE for all sample sizes for P < 0.9; for P > 0.99 the POME had the
least RMSE. Thus, it is inferred that the MOM, PWM or MLE may be used
for smaller values of G, but for higher values of G, the POME may be the
preferred method.
C O N C L U S I O N S
The following conclusions can be drawn from this study: (1) the POME offers
an alternative method for estimating the parameters of the 3-parameter
generalized Pareto distribution; (2) when the skewness was high (G = 2.5), the
PO M E yielded superior param eter estimates; (3) for low skewness (G = 0.5 ),
the POME was better in parameter estimates than the MLE and PWM but
worse than the MOM; however, for large sample size, its performance
improved significantly; (4) the PO M E produced either better or comparable
quantile estimates as compared with the MOM, MLE and PWM for high
skewness (G = 2.5); (5) for low skewness (G = 0.5), the POME was
comparable to the MOM, the MLE and the PWM for lower probabilities of
nonexceedance which for higher values, the MOM or PWM was better than the
POME.
R E F E R E N C E S
Barrett, J. H. (1992) An extreme value analysis of the flow of Burbage Broo k. Stochastic Hydrol. Hydraul.
6, 151-165.
Baxter, M . A. (1980) Minimu m variance unbiased estimation of the param eter of the Pareto distribution.
Biometrika 27, 133-138.
Cook, W. L. & Mumme, D. C. (1981) Estimation of Pareto parameters by numerical methods. In:
Statistical Distributions in Scientific Work, d. C. Taillie et al. 5, 127-132.
Davison, A. C. (1984a) Modelling excesses over high thresholds, with an application. In: Statistical
Extremes and Applications, ed. J. Tiago de Oliveira, 46 1-482 . Reidel, Dordrecht, The Netherlands.
Dav ison, A. C . (1984b) A statistical model for contamination due to long-range atmosph eric transpo rt of
radionuclides. PhD thesis, Department of Mathematics, Imperial College of Science and
Technology, London, UK.
Davison, A. C . & Smith, R. L. (1990) Models for exceedances over high thresholds. /. Roy. Statist. Soc.
B 52(3), 393-442.
8/9/2019 Parameter Estimation for 3-Parameter Pareto
17/18
Parameterestimation forgeneralized Pareto distribution 181
Du Mo uche l, W . (1983) Estimating the stable indexa in order to measure tail thickness.Ann. Statist. 11,
1019-1036.
Ho sking , J. R. M . & W allis, J. R. (1987) Parameter and quantile estimation for the generalized Pareto
distribution.
Technometrics
29(3), 339-349.
Jaynes, E. T. (1961) Probability Theory in Science and Engineering. McGraw-Hill, New York, USA.
Jayn es, E. T. (1968) Prior probabilities. IEEE Trans. Syst. Man. Cybern.3(SSC-4),
227-241.
Jenkin son, A. F . (1955) The frequency distribution of the annual maximum (or minimum) of meteorological
elements. Quart. J. Roy. Meteorol. Soc. 8 1 ,158-171.
Jin, M . & S tedinger, J. R. (1989) Partial duration series analysis for a GEV annual flood distribution with
systematic and historical flood information (unpublished paper). Department of Civil Engineering,
Pennsylvania State University, State College, PA, USA.
Joe, H . (198 7) Estimation of quantiles of the maximum of N observations.Biometrika 74, 347-354.
Levine, R. D. & Tribus, M. (1979) The Maximum Entropy Formalism. MIT Press, Cambridge,
Massachusetts, USA.
Picka nds, J. (1975 ) Statistical inference using extreme order statistics. Ann. Statist. 3 , 119-131.
Quan dt, R. E. (1966) Old and new methods of estimation of the Pareto distribution.Biometrika 10, 55-82.
Rosb jerg, D. , Madse n, H. & Rasmussen, P. F. (1992) Prediction in partial duration series with generalized
Pareto-distributed exceedances.Wat. Resour. Res. 28(11), 3001-3010.
Saksena, S. K . & Johnson , A. M. (1984) Best unbiased estimators for the parameters of a two-param eter
Pareto distribution. Biometrika
3 1 ,
77-83.
Shanno n, C. E . (1948) The mathematical theory of communication, I-IV.Bell System Tech. J. 27, 279-428,
612-656.
Singh, V. P. & Fioren tino, M. (1992) A historical perspective of entropy applications in water resourc es.
In :Entropy and Energy D issipation in Wa ter Resources, ed. V. P. Singh & M. Fiorentino,
21-61.
Kluwer, Dordrecht, The Netherlands.
Singh, V. P. & Rajagopal, A. K . (1986) A new method of parameter estimation for hydrologie frequency
analysis. Hydrol. Sci. Technol. 2(3), 33-40.
Smith, J. A . (1991 ) Estimating the upp er tail of flood frequency distributions. Wat. Resour. Res. 23(8),
1657-1666.
Smith, R. L. (1984) Threshold methods for sample extreme s, In: Statistical Extremes and App lications ed.
J. Trago de Oliveira, 6 21-638. Reidel, Dordrecht, The Netherlands.
Smith, R. L . (1987) Estimating tails of probability distributions. Ann. Statist., 15, 1174-1207.
Tribus, M. (1969) Rational D escriptions, D ecisions and D esigns. Pergamon, New York, USA.
van Montfort, M. A. J. & Witter, J. V. (1985) Testing exponentiality against generalized Pareto
distribution. /. Hydrol. 78, 305-315.
van Montfort, M. A. J. & Witter, J. V. (1986) The generalized pareto distribution applied to rainfall
depths.
Hydrol. Sci. J.
31(2), 151-162.
van Montfort, M . A. J. & Otten, A. (1991) The first and the second e of the extreme value distribution,
E V 1 . Stochastic Hydrol. Hydraul. 5, 69-76.
W ang, Q. J. (1990) Studies on statistical methods of flood frequency analysis. PhD dissertation, National
University of Ireland, Galway, Ireland.
W ang, Q . J. (1991) The PO T model described by the generalized Pareto distribution with Poisson arrival
rate. X
Hydrol.
129,263-280.
Received 8 February 1993; 22 September 1994
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