Reliability Analysis of the Proportional Mean Residual Life Order

Research ArticleReliability Analysis of the Proportional Mean ResidualLife Order

M Kayid12 S Izadkhah3 and H Alhalees1

1 Department of Statistics and Operations Research College of Science King Saud University Riyadh 11451 Saudi Arabia2Department of Mathematics Faculty of Science Suez University Suez 41522 Egypt3 School of Mathematical Sciences Ferdowsi University of Mashhad Mashhad 91779 Iran

Correspondence should be addressed to M Kayid el kayid2000yahoocom

Received 4 April 2014 Accepted 2 August 2014 Published 28 August 2014

Academic Editor Shaomin Wu

Copyright copy 2014 M Kayid et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The concept of mean residual life plays an important role in reliability and life testing In this paper we introduce and study a newstochastic order called proportional mean residual life order Several characterizations and preservation properties of the new orderunder some reliability operations are discussed As a consequence a new class of life distributions is introduced on the basis of theanti-star-shaped property of the mean residual life function We study some reliability properties and some characterizations ofthis class and provide some examples of interest in reliability

1 Introduction

Stochastic orders have shown that they are very useful inapplied probability statistics reliability operation researcheconomics and other related fields Various types of stochas-tic orders and associated properties have been developedrapidly over the years Let 119883 be a nonnegative random vari-able which denotes the lifetime of a system with distributionfunction 119865 survival function 119865 = 1minus119865 and density function119891 The conditional random variable119883

119905= (119883minus119905 | 119883 gt 119905) 119905 ge

0 is known as the residual life of the system after 119905 given thatit has already survived up to 119905 The mean residual life (MRL)function of119883 is the expectation of119883

119905 which is given by

120583119883(119905) =

int

infin

119905

119865 (119906)

119865 (119905)

119889119906 119905 gt 0

0 119905 le 0

(1)

The MRL function is an important characteristic invarious fields such as reliability engineering survival analysisand actuarial studies It has been extensively studied in theliterature especially for binary systems that is when there areonly two possible states for the system as either working or

failed Another useful reliability measure is the hazard rate(HR) function of119883 which is given by

119903119883(119905) =

119891 (119905)

119865 (119905)

119905 ge 0 (2)

TheHR function is particularly useful in determining theappropriate failure distributions utilizing qualitative informa-tion about the mechanism of failure and for describing theway in which the chance of experiencing the event changeswith time In replacement and repair strategies althoughthe shape of the HR function plays an important role theMRL function is found to be more relevant than the HRfunction because the former summarizes the entire residuallife function whereas the latter involves only the risk ofinstantaneous failure at some time 119905 For an exhaustivemono-graph on the MRL and HR functions and their reliabilityanalysis we refer the readers to Ramos-Romero and Sordo-Dıaz [1] Belzunce et al [2] and Lai and Xie [3] Based on theMRL function a well-knownMRLorder has been introducedand studied in the literature Gupta and Kirmani [4] andAlzaid [5] were among the first who proposed theMRL orderOver the years many authors have investigated reliabilityproperties and applications of the MRL order in reliabilityand survival analysis (cf Shaked and Shanthikumar [6] and

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 142169 8 pageshttpdxdoiorg1011552014142169

2 Mathematical Problems in Engineering

Muller and Stoyan [7]) On the other hand the proportionalstochastic orders are considered in the literature to generalizesome existing notions of stochastic comparisons of randomvariables Proportional stochastic orders as extended versionsof the existing common stochastic orders in the literaturewere studied by some researchers such asRamos-Romero andSordo-Dıaz [1] and Belzunce et al [2] Recently Nanda et al[8] gave an effective review of the different partial orderingresults related to the MRL order and studied some reliabilitymodels in terms of the MRL function

The purpose of this paper is to propose a new stochasticorder called proportional mean residual life (PMRL) orderwhich extends the MRL order to a more general settingSome implications characterization properties and preser-vation results under weighted distributions of this new orderincluding its relationships with other well-known orders arederived In addition two characterizations of this order basedon residual life at random time and the excess lifetime inrenewal processes are obtained As a consequence a newclass of lifetime distributions namely anti-star-shaped meanresidual life (ASMRL) class of life distribution which isclosely related to the concept of the PMRL order is intro-duced and studied A number of useful implications char-acterizations and examples for this class of life distributionsare discussed along with some reliability applications Thepaper is organized as followsThe precise definitions of somestochastic orders as well as some classes of life distributionswhich will be used in the sequel are given in Section 2 Inthat section the PMRL order is introduced and studiedSeveral characterizations and preservation properties of thisnew order under some reliability operations are discussedIn addition to illustrate the concepts some applications inthe context of reliability theory are included In Section 3 theASMRL class of life distributions is introduced and studiedFinally in Section 4 we give a brief conclusion and someremarks of the current research and its future

Throughout this paper the term increasing is used insteadof monotone nondecreasing and the term decreasing is usedinstead of monotone nonincreasing Let us consider tworandom variables 119883 and 119884 having distribution functions119865 and 119866 respectively and denote by 119865(119891) and 119866(119892) theirrespective survival (density) functions We also assume thatall random variables under consideration are absolutelycontinuous and have 0 as the common left endpoint of theirsupports and all expectations are implicitly assumed to befinite whenever they appear In addition we use the notationsR = (minusinfininfin) R+ = (0infin)

119904119905

= denotes the equality indistribution and 119883V is the weighted version of 119883 accordingto the weight V

2 Proportional Mean Residual Life Order

For ease of reference before stating our main results let usrecall some stochastic orders classes of life distributions anddependence concepts which will be used in the sequel

Definition 1 The random variable119883 is said to be smaller than119884 in the

(i) HR order (denoted as119883leHR119884) if

119866 (119905)

119865 (119905)

is increasing in 119905 isin R+ (3)

(ii) reversed hazard (RH) order (denoted as119883leRH119884) if

119891 (119909)

119865 (119909)le119892 (119909)

119866 (119909) forall119909 isin R

+ (4)

which denotes the reversed hazard (RH) rate order(iii) MRL order (denoted as119883leMRL119884) if

intinfin

119905119866 (119906) 119889119906

intinfin

119905119865 (119906) 119889119906

is increasing in 119905 isin R+ (5)

Definition 2 (Lai and Xie [3]) The nonnegative randomvariable 119883 is said to have a decreasing mean residual life(DMRL) whenever the MRL of119883 is decreasing

Definition 3 (Lariviere and Porteus [9]) The nonnegativerandom variable 119883 is said to have an increasing generalizedfailure rate (IGFR) whenever the generalized failure ratefunction 120573

119883of 119883 which is given by 120573

119883(119909) = 119909119903

119883(119909) is

increasing in 119909 ge 0Note that in view of a result in Lariviere [10] 119883 has

IGFR property if and only if 119909119883leHR119883 for all 119909 isin (0 1] orequivalently if119883leHR119909119883 for any 119909 ge 1

Definition 4 (Karlin [11]) A nonnegative measurable func-tion ℎ(119909 119910) is said to be totally positive of order 2 (TP

2) in

119909 isin R and 119910 isin R whenever

10038161003816100381610038161003816100381610038161003816

ℎ (1199091 1199101) ℎ (119909

1 1199102)

ℎ (1199092 1199101) (119909

2 1199102)

10038161003816100381610038161003816100381610038161003816

ge 0 forall1199091le 1199092 1199101le 1199102 (6)

Definition 5 (Shaked and Shanthikumar [6]) A nonnegativefunction 120595 is said to be anti-star-shaped on a set 119860 sube R+

if 120595(120572119909) ge 120572120595(119909) for all 119909 on 119860 and for every 120572 isin

[0 1] Equivalently 120595 is anti-star-shaped on 119860 if 120595(119909)119909 isnonincreasing in 119909 isin 119860

Below we present the definition of the proportionalhazard rate (PHR) order and its related proportional agingclass

Definition 6 (see Belzunce et al [2]) Let 119883 and 119884 be twononnegative continuous random variables It is said that

(i) 119883 is smaller than 119884 in the PHR order (denoted as119883leP-HR119884) if 119909119883leHR119884 for all 119909 isin (0 1]

(ii) 119883 is increasing proportional hazard rate (IPHR) if119909119883leHR119883 for all 119909 isin (0 1]

Consider the situation wherein 119883 denotes the risk thatthe direct insurer faces and 120601 the corresponding reinsurancecontract One important reinsurance agreement is quota-share treaty defined as 120601(119883) = 119909119883 for 119909 isin (0 1]The randomvariable 119884 denotes the risk that an independent insurer

Mathematical Problems in Engineering 3

faces Insurers sometimes seek a quota-share treaty whenthey require financial support from their reinsurers thusmaintaining an adequate relation between net income andcapital reserves Motivated by this we propose the followingnew stochastic order

Definition 7 Let 119883 and 119884 be two nonnegative randomvariables The random variable 119883 is smaller than 119884 in thePMRL order (denoted as 119883leP-MRL119884) if 119909119883leMRL119884 for all119909 isin (0 1]

Remark 8 Note that 119883leP-MRL119884 hArr 119909119883leMRL119884 for all 119909 isin(0 1] thus with 119909 = 1 we have 119883leMRL119884 that is the PMRLorder is stronger than the MRL order

The first results of this section provide an equivalentcondition for the PMRL order

Theorem 9 The following assertions are equivalent

(i) 119883leP-MRL119884(ii) 119909120583

119883(119905) le 120583

119884(119909119905) for all 119905 ge 0 and each 119909 isin (0 1]

(iii) intinfin119909119905119866(119906)119889119906 int

infin

119905119865(119906)119889119906 is increasing in 119905 for all 119909 isin

(0 1]

Proof First we prove that (i) and (ii) are equivalent Note thatthe MRL of 119909119883 as a function of 119904 is given by 119909120583

119883(119904119909) for all

119904 ge 0 and for any 119909 isin (0 1] Now we have 119883leP-MRL119884 if forall 119909 isin (0 1] it holds that

119909119883leMRL119884

lArrrArr 119909120583119883(119904

119909) le 120583119884(119904) forall119904 ge 0

lArrrArr 119909120583119883(119905) le 120583

119884(119909119905) forall119905 ge 0

where 119905 = 119904

119909

(7)

To prove that (ii) and (iii) are equivalent we have

119889

119889119905

intinfin

119909119905119866 (119906) 119889119906

intinfin

119905119865 (119906) 119889119906

=

119865 (119905) intinfin

119909119905119866 (119906) 119889119906 minus 119909119866 (119909119905) int

infin

119905119865 (119906) 119889119906

[intinfin

119905119865 (119906) 119889119906]

2

=119865 (119905) 119866 (119909119905)

[intinfin

119905119865(119906)119889119906]

2times [120583119884(119909119905) minus 119909120583

119883(119905)]

(8)

It is obvious that the last term is nonnegative if and only if119909120583119883(119905) le 120583

119884(119909119905) for all 119905 ge 0 and for any 119909 isin (0 1]

In the context of reliability engineering and survivalanalysis weighted distributions are of tremendous practicalimportance (cf Jain et al [12] Bartoszewicz and Skolimowska[13] Misra et al [14] Izadkhah and Kayid [15] and Kayid etal [16]) In renewal theory the residual lifetime has a limitingdistribution that is a weighted distribution with the weight

function equal to the reciprocal of the HR function Someof the well-known and important distributions in statisticsand applied probability may be expressed as weighted dis-tributions such as truncated distributions the equilibriumrenewal distribution distributions of order statistics anddistributions arisen in proportional hazards and proportionalreversed hazards models Recently Izadkhah et al [17] haveconsidered the preservation property of the MRL orderunder weighted distributions Here we develop a similarpreservation property for the PMRL order under weighteddistributions For two weight functions 119908

1and 119908

2 assume

that1198831199081

and1198841199082

denote the weighted versions of the randomvariables of 119883 and 119884 respectively with respective densityfunctions

1198911(119909) =

1199081(119909) 119891 (119909)

]1

1198921(119909) =

1199082(119909) 119892 (119909)

]2

(9)

where 0 lt ]1= 119864(119908

1(119883)) lt infin and 0 lt ]

2= 119864(119908

2(119884)) lt infin

Let 1198611(119909) = 119864(119908

1(119883) | 119883 gt 119909) and 119861

2(119909) = 119864(119908

2(119884) | 119884 gt

119909) Then survival functions of 1198831199081

and 1198841199082

are respectivelygiven by

1198651(119909) =

1198611(119909) 119865 (119909)

]1

1198661(119909) =

1198612(119909) 119866 (119909)

]2

(10)

First we consider the following useful lemma which isstraightforward and hence the proof is omitted

Lemma 10 Let 119883 be a nonnegative absolutely continuousrandom variable Then for any weight function 119908

1

(119909119883)V119904119905

= 1199091198831199081

forall119909 gt 0 (11)

where V is a weight function of the form V(119905) = 1199081(119905119909)

Theorem 11 Let 1198612

be an increasing function and let1198612(119904119909)119861

1(119904) increase in 119904 ge 0 for all 119909 isin (0 1] Then

119883leP-MRL119884 997904rArr 1198831199081

leP-MRL1198841199082

(12)

Proof Let 119909 isin (0 1] be fixed Then 119883leP-MRL119884 gives119909119883leMRL119884 We know by assumption that 119861

2is increasing and

the ratio

119864 (1199082(119884) | 119884 gt 119905)

119864 (V (119909119883) | 119909119883 gt 119905)

=119864 (1199082(119884) | 119884 gt 119905)

119864 (1199081(119883) | 119883 gt 119905119909)

=119864 (1199082(119884) | 119884 gt 119904119909)

119864 (1199081(119883) | 119883 gt 119904)

=1198612(119904119909)

1198611(119904)

(13)

is increasing in 119905 ge 0 when 119904 = 119905119909 In view of Theorem 2 inIzadkhah et al [17] we conclude that (119909119883)VleMRL119884119908

2

Becauseof Lemma 10 and because the equality in distribution of (119909119883)Vand 119909119883

1199081

implies the equality in their MRL functions itfollows that 119909119883

1199081

leMRL1198841199082

So for all 119909 isin (0 1] we have1199091198831199081

leMRL1198841199082

which means that1198831199081

leP-MRL1198841199082

4 Mathematical Problems in Engineering

On the other hand in many reliability engineering prob-lems it is interesting to study 119883

119884= [119883 minus 119884 | 119883 gt 119884] the

residual life of 119883 with a random age 119884 The residual life atrandom time (RLRT) represents the actual working time ofthe standby unit if 119883 is regarded as the total random life ofa warm standby unit with its age 119884 For more details aboutRLRT we refer the readers to Yue and Cao [18] Li and Zuo[19] and Misra et al [20] among others Suppose that119883 and119884 are independentThen the survival function of119883

119884 for any

119909 ge 0 is given by

119875 (119883119884gt 119909) =

intinfin

0119865 (119909 + 119910) 119889119866 (119910)

intinfin

0119865 (119910) 119889119866 (119910)

(14)

Theorem 12 Let 119883 and 119884 be two nonnegative randomvariables 119883

119884leP-MRL119883 for any 119884 which is independent of 119883 if

and only if

119883119905leP-MRL119883 forall119905 ge 0 (15)

Proof To prove the ldquoif rdquo part let 119883119905leP-MRL119883 for all 119905 ge 0 It

then follows that for all 119904 gt 0 and 119909 isin (0 1]

int

infin

119904

119909119865 (119905 + 119906) 119889119906 le119865 (119905 + 119904)

119865 (119904119909)

int

infin

119904119909

119865 (119906) 119889119906 (16)

By integrating both sides of (16) with respect to 119905 through themeasure 119866 we have

int

infin

0

int

infin

119904

119909119865 (119905 + 119906) 119889119906 119889119866 (119905)

le int

infin

0

[119865 (119905 + 119904)

119865 (119904119909)

int

infin

119904119909

119865 (119906) 119889119906] 119889119866 (119905)

= int

infin

0

119865 (119905 + 119904) 119889119866 (119905)

intinfin

119904119909119865 (119906) 119889119906

119865 (119904119909)

forall119904 ge 0 119909 isin (0 1]

(17)

which is equivalent to saying that119883119884leP-MRL119883 for all 119884rsquos that

are independent of 119883 For the ldquoonly if rdquo part suppose that119883119884leP-MRL119883 holds for any nonnegative random variable 119884

Then 119883119905leP-MRL119883 for all 119905 ge 0 follows by taking 119884 as a

degenerate random variable

Let 119883119899 119899 = 1 2 be a sequence of mutually

independent and identically distributed (iid) nonnegativerandom variables with common distribution function 119865 For119899 ge 1 denote 119878

119899= sum119899

119894=1119883119894which is the time of the 119899th

arrival and 1198780= 0 and let 119873(119905) =Sup119899 119878

119899le 119905 represent

the number of arrivals during the interval [0 119905] Then 119873 =

119873(119905) 119905 ge 0 is a renewal processwith underlying distribution119865 (see Ross [21]) Let 120574(119905) be the excess lifetime at time 119905 ge 0that is 120574(119905) = 119878

119873(119905)+1minus119905 In this context we denote the renewal

function by 119872(119905) = 119864[119873(119905)] which satisfies the followingwell-known fundamental renewal equation

119872(119905) = 119865 (119905) + int

119905

0

119865 (119905 minus 119910) 119889119872(119910) 119905 ge 0 (18)

According to Barlow and Proschan [22] it holds for all 119905 ge 0and 119909 ge 0 that

119875 (120574 (119905) gt 119909) = 119865 (119905 + 119909) + int

119905

0

119865 (119905 + 119909 minus 119911) 119889119872 (119911) (19)

In the literature several results have been given tocharacterize the stochastic orders by the excess lifetime ina renewal process Next we investigate the behavior of theexcess lifetime of a renewal process with respect to the PMRLorder

Theorem 13 If 119883119905leP-MRL119883 for all 119905 ge 0 then 120574(119905)leP-MRL120574(0)

for all 119905 ge 0

Proof First note that 119883119905leP-MRL119883 for all 119905 ge 0 if and only if

for any 119905 ge 0 119904 gt 0 and 119909 isin (0 1]

int

infin

119904

119909119865 (119905 + 119906) 119889119906 le 119865 (119905 + 119904)

intinfin

119904119909119865 (119906) 119889119906

119865 (119904119909)

(20)

In view of the identity of (19) and the inequality in (20) wecan get

int

infin

119904

119909119875 (120574 (119905) gt 119906) 119889119906

= int

infin

119904

119909119865 (119905 + 119906) 119889119906

+ int

119905

0

int

infin

119904

119909119865 (119905 minus 119910 + 119906) 119889119906 119889119872(119910)

le int

infin

119904

119909119865 (119905 + 119906) 119889119906

+ int

119905

0

[119865 (119905 minus 119910 + 119904)

119865 (119904119909)

int

infin

119904119909

119865 (119906) 119889119906] 119889119872(119910)

= int

infin

119904

119909119865 (119905 + 119906) 119889119906

+

intinfin

119904119909119865 (119906) 119889119906

119865 (119904119909)

[119875 (120574 (119905) gt 119904) minus 119865 (119905 + 119904)]

le

intinfin

119904119909119865 (119906) 119889119909

119865 (119904119909)

[119865 (119905 + 119904)]

+

intinfin

119904119909119865 (119906) 119889119906

119865 (119904119909)

[119875 (120574 (119905) gt 119904) minus 119865 (119905 + 119904)]

=

intinfin

119904119909119865 (119906) 119889119906

119865 (119904119909)

119875 (120574 (119905) gt 119904)

(21)

Hence it holds that for all 119905 ge 0 119904 gt 0 and for any 119909 isin (0 1]

intinfin

119904119909119875 (120574 (119905) gt 119906) 119889119906

119875 (120574 (119905) gt 119904)le

intinfin

119904119909119865 (119906) 119889119906

119865 (119904119909)

(22)

which means 120574(119905)leP-MRL 120574(0) for all 119905 ge 0

Mathematical Problems in Engineering 5

3 Anti-Star-Shaped Mean Residual Life Class

Statisticians and reliability analysts have shown a growinginterest in modeling survival data using classifications of lifedistributions by means of various stochastic orders Thesecategories are useful for modeling situations maintenanceinventory theory and biometry In this section we proposea new class of life distributions which is related to the MRLfunction We study some characterizations preservationsand applications of this new class Some examples of interestin the context of reliability engineering and survival analysisare also presented

Definition 14 The lifetime variable 119883 is said to have an anti-star-shapedmean residual life (ASMRL) if theMRL functionof119883 is anti-star-shaped

It is simply derived that 119883 isin ASMRL whenever 120583119883(119905)119905

is decreasing in 119905 gt 0 Useful description and motivationfor the definition of the ASMRL class which is due toNanda et al [8] are the following Consider a situation inwhich 119883 represents the risk that the direct insurer facesand 120601 the corresponding reinsurance contract The ASMRLclass provides that the quota-share treaty related to a riskis less than risk itself in the sense of the MRL order Inwhat follows we focus on the ASMRL class as a weakerclass than the DMRL class to get some basic results Firstconsider the following characterization property which canbe immediately obtained byTheorem 9(ii)

Theorem 15 The lifetime random variable119883 is ASMRL if andonly if 119883leP-MRL119883

Theorem 16 The lifetime random variable119883 is ASMRL if andonly if

1205791119883leMRL1205792119883 for any 120579

1le 1205792isin R+ (23)

Proof Denote119883(120579119894) = 120579119894119883 for 119894 = 1 2 TheMRL function of

119883(120579119894) is then given by 120583

119883(120579119894)(119905) = 120579

119894120583119883(119905120579119894) for all 119905 ge 0 and

119894 = 1 2In view of the fact that 1205791119883leMRL1205792119883 for all 1205791 le 1205792 isin

R+ if and only if

1205791120583119883(119905

1205791

) le 1205792120583119883(119905

1205792

) for any 1205791le 1205792isin R+ (24)

By taking 119909 = 12057911205792and 119904 = 119905120579

1the above inequality is

equivalent to saying that 119909120583119883(119904) le 120583

119883(119909119904) for all 119904 ge 0 and

for any 119909 isin (0 1] This means that119883 is ASMRL

Remark 17 The result ofTheorem 16 indicates that the familyof distributions 119865

120579(119909) = 119865(119909120579) 120579 gt 0 is stochastically

increasing in 120579 with respect to the MRL order if and onlyif the distribution 119865 has an anti-star-shaped MRL functionAnother conclusion ofTheorem 16 is to say that119883leP-MRL119883 ifand only if119883leMRL119909119883 for all 119909 isin [1infin)

Theorem 18 If119883leMRL119884 and if either119883 or 119884 has an anti-star-shaped MRL function then119883leP-MRL119884

Proof Let119883leMRL119884 and let119883 be ASMRL Then we have

119909119883leMRL119883leMRL119884 forall119909 isin (0 1] (25)

Hence it holds that 119909119883leMRL119884 for all 119909 isin (0 1] which means119883leP-MRL119884Theproof of the result when119884 is ASMRL is similarby taking the fact that 119884 is ASMRL if and only if 119884leMRL120572119884for all 120572 ge 1 into account Note also that 119883leP-MRL119884 if andonly if119883leMRL120572119884 for any 120572 ge 1

The following counterexample shows that the MRL orderdoes not generally imply the ASMRL order and hence thesufficient condition inTheorem 18 cannot be removed

Counter Example 1 Let 119883 have MRL 120583119883(119905) = (119905 minus 12)

2 for119905 isin [0infin) and let 119884 have MRL 120583

119884(119905) = 16 for 119905 isin [0 16]

and 120583119884(119905) = 3(119905 minus 12)

22 for 119905 isin (16infin) These MRL

functions are readily shown not to be ASMRL We can alsosee that 120583

119883(119905) le 120583

119884(119905) for all 119905 ge 0 that is119883leMRL119884 It can be

easily checked now that 119909119883≰MRL119884 for 119909 = 14 which meansthat119883≰P-MRL119884

As an obvious conclusion of Theorem 18 above andTheorem 29 in Nanda et al [8] if 119883 is DMRL then 119883 isASMRL The next result presents another characterization ofthe ASMRL class

Theorem 19 A lifetime random variable 119883 is ASMRL if andonly if119885119883leMRL119883 for each random variable119885with 119878

119885= (0 1]

which is independent of119883

Proof To prove the ldquoif rdquo part note that 120583119911119883(119905) = 119911120583

119883(119905119911)

for each 119911 isin (0 1] and any 119905 gt 0 Take 119885 = 119911 for each 119911 isin(0 1] one at a time as a degenerate random variable implying119911119883leMRL119883 for all 119911 isin (0 1] which means 119883 isin ASMRL Forthe ldquoonly if rdquo part assume that 119885 has distribution function 119866From the assumption and the well-known Fubini theoremfor all 119909 gt 0 it follows that

120583119885119883(119909) =

intinfin

119909119875 (119885119883 gt 119906) 119889119906

119875 (119885119883 gt 119909)

=

intinfin

119909int1

0119865 (119906119911) 119889119866 (119911) 119889119906

int1

0119865 (119909119911) 119889119866 (119911)

=

int1

0intinfin

119909119865 (119906119911) 119889119906 119889119866 (119911)

int1

0119865 (119909119911) 119889119866 (119911)

=

int1

0119911119898119883(119909119911) 119865 (119909119911) 119889119866 (119911)

int1

0119865 (119909119911) 119889119866 (119911)

le

int1

0119898119883(119909) 119865 (119909119911) 119889119866 (119911)

int1

0119865 (119909119911) 119889119866 (119911)

= 120583119883(119909)

(26)

That is 119885119883leMRL119883

The following result presents a sufficient condition for aprobability distribution to be ASMRL

6 Mathematical Problems in Engineering

Theorem 20 Let the lifetime random variable 119883 be IGFRThen119883 is ASMRL

Proof Recall that119883 is IGFR if and only if 119909119903119883(119909) is increasing

in 119909 ge 0 Because of the identity

119905119865 (119905) = int

infin

119905

119906119891 (119906) 119889119906 minus int

infin

119905

119865 (119906) 119889119906 forall119905 gt 0 (27)

we can write for all 119905 gt 0

120583119883(119905)

119905=

intinfin

119905119865 (119906) 119889119906

119905119865 (119905)

=

intinfin

119905119865 (119906) 119889119906

intinfin

119905119906119891 (119906) 119889119906 minus int

infin

119905119865 (119906) 119889119906

=1

(intinfin

119905119906119891 (119906) 119889119906 int

infin

119905119865 (119906) 119889119906) minus 1

(28)

Thus 120583119883(119905)119905 is decreasing in 119905 gt 0 if and only if the ratio

intinfin

119905119906119891 (119906) 119889119906

intinfin

119905119865 (119906) 119889119906

is increasing in 119905 gt 0 (29)

Let

120588 (119894 119905) = int

infin

0

120601 (119894 119906) 120595 (119906 119905) 119889119906 (30)

as a function of 119894 = 1 2 and of 119905 gt 0 where

120601 (119894 119906) = 119906119891 (119906) if 119894 = 2119865 (119906) if 119894 = 1

120595 (119906 119905) = 1 if 119906 gt 1199050 if 119906 le 119905

(31)

Note that the ratio given in (29) is increasing in 119905 gt 0

if and only if 120588 is TP2in (119894 119905) isin 1 2 times (0infin) From

the assumption since 119906119903119883(119906) is increasing then 120601 is TP

2in

(119894 119906) isin 1 2 times (0infin) Also it is easy to see that 120595 is TP2in

(119906 119905) isin (0infin) times (0infin) By applying the general compositiontheorem of Karlin [11] to the equality of (30) the proof iscomplete

To demonstrate the usefulness of the ASMRL class inreliability engineering problems we consider the followingexamples

Example 21 The Weibull distribution is one of the mostwidely used lifetime distributions in reliability engineering Itis a versatile distribution that can take on the characteristicsof other types of distributions based on the value of the shapeparameter Let 119883 have the Weibull distribution with survivalfunction

119865 (119909) = exp (minus(120582119909)120572) 119909 ge 0 120572 gt 0 120582 gt 0 (32)

The HR function is given by 119903119883(119909) = 120572120582

120572119909120572minus1 Thus we

have 119909119903119883(119909) = 120572(120582119909)

120572 which is increasing in 119909 gt 0 forall parameter values and hence according toTheorem 20119883 isASMRL

Example 22 The generalized Pareto distribution has beenextensively used in reliability studies when robustness isrequired against heavier tailed or lighter tailed alternativesto an exponential distribution Let119883 have generalized Paretodistribution with survival function

119865 (119909) = (119887

119886119909 + 119887)

(1119886)+1

119909 ge 0 119886 gt 0 119887 gt 0 (33)

The HR function is given by 119903119883(119909) = (1 + 119886)(119886119909 + 119887) Thus

we get 119909119903119883(119909) = (1 + 119886)119909(119886119909 + 119887) which is increasing in 119909

for all parameter values and soTheorem 20 concludes that119883is ASMRL

Example 23 Let 119879119894be a lifetime variable having survival

function given by 119865119894(119905) = 119864(119866(119885

119894119905)) 119905 ge 0 where 119885

119894is a

nonnegative random variable and 119866 is the survival functionof a lifetime variable 119884 for each 119894 = 1 2 This is called scalechange random effects model in Ling et al [23] Noting thefact that 119884leMRL119896119884 for all 119896 ge 1 is equivalent to saying that119884 is ASMRL according to Theorem 310 of Ling et al [23] if1198851leRH1198852 and 119884 is ASMRL then 119879

1geMRL1198792

In the context of reliability theory shock models areof great interest The system is assumed to have an abilityto withstand a random number of these shocks and it iscommonly assumed that the number of shocks and theinterarrival times of shocks are s-independent Let119873 denotethe number of shocks survived by the system and let 119883

119895

denote the random interarrival time between the (119895 minus 1)thand 119895th shocks Then the lifetime 119879 of the system is givenby 119879 = sum

119873

119895=1119883119895 Therefore shock models are particular

cases of random sums In particular if the interarrivals areassumed to be s-independent and exponentially distributed(with common parameter 120582) then the distribution functionof 119879 can be written as

119867(119905) =

infin

sum

119896=0

119890minus120582119905(120582119905)119896

119896119875119896 119905 ge 0 (34)

where 119875119896= 119875[119873 le 119896] for all 119896 isin 119873 (and 119875

0= 1) Shock

models of this kind called Poisson shock models have beenstudied extensively For more details we refer to Fagiuoli andPellerey [24] Shaked andWong [25] Belzunce et al [26] andKayid and Izadkhah [27]

In the following we make conditions on the randomnumber of shocks under which119879 has ASMRL property Firstlet us define the discrete version of the ASMRL class

Definition 24 A discrete distribution 119875119896is said to have

discrete anti-star-shaped mean residual life (D-ASMRL)property if suminfin

119895=119896119875119895119896119875119896minus1

is nonincreasing in 119896 isin 119873

Theorem 25 If 119875119896 119896 isin 119873 in (34) is D-ASMRL then 119879 with

the sf119867(119905) as given in (34) is ASMRL

Mathematical Problems in Engineering 7

Proof Wemay note that for all 119905 gt 0

119905119867 (119905) =1

120582

infin

sum

119896=0

119890minus120582119905 (120582119905)

119896

119896119896119875119896minus1

int

infin

119905

119867(119906) 119889119906 =1

120582

infin

sum

119896=0

119890minus120582119905 (120582119905)

119896

119896

infin

sum

119895=119896

119875119895

(35)

Hence 119879 is ASMRL if and only if

119905119867 (119905)

intinfin

119905119867(119906) 119889119906

=

suminfin

119896=0119890minus120582119905((120582119905)119896119896) 119896119875

119896minus1

suminfin

119896=0119890minus120582119905 ((120582119905)

119896119896)sum

infin

119895=119896119875119895

(36)

is increasing in 119905 or equivalently if

Ψ (119894 119905) =

infin

sum

119896=0

Φ (119894 119896)(120582119905)119896

119896119890minus120582119905 is TP

2in (119894 119905) (37)

for 119894 isin 1 2 and 119905 isin R+ where

Φ (119894 119896) =

infin

sum

119895=119896

119875119895 if 119894 = 1

119896119875119896minus1 if 119894 = 2

(38)

By the assumption Φ(119894 119896) is TP2in (119894 119896) for 119894 isin 1 2 and

119896 isin 119873 It is also evident that 119890minus120582119905(120582119905)119896119896 is TP2in (119896 119905) for

119896 isin 119873 and 119905 isin R+ The result now follows from the generalcomposition theorem of Karlin [11]

Lemma 26 Let 1198831 1198832 119883

119899be an iid sample from

119865 and let 1198841 1198842 119884

119899be an iid sample from 119866 Then

min1198831 1198832 119883

119899leMRL min119884

1 1198842 119884

119899 implies

119883119894leMRL119884119894 119894 = 1 2 119899

Example 27 Reliability engineers often need to workwith systems having elements connected in seriesLet 119883

1 1198832 119883

119899be iid random lifetimes such that

119879 = min1198831 1198832 119883

119899 has the ASMRL property Then

according to Theorem 15 119896119879leMRL119879 for all 119896 isin (0 1] Thismeans that

min 1198961198831 1198961198832 119896119883

119899

leMRL min 1198831 1198832 119883

119899 forall119896 isin (0 1]

(39)

By appealing to Lemma 26 it follows that 119896119883119894leMRL119883119894 119894 =

1 2 119899 for all 119896 isin (0 1] That is 119883119894 119894 = 1 2 119899

is ASMRL Hence the ASMRL property passes from thelifetime of the series system to the lifetime of its iidcomponents

Accelerated life models relate the lifetime distribution tothe explanatory variables (stress covariates and regressor)This distribution can be defined by the survival cumulativedistribution or probability density functions Neverthelessthe sense of accelerated life models is best seen if they areformulated in terms of the hazard rate function In thefollowing example we state an application of Theorem 16 inaccelerated life models

Example 28 Consider 119899 units (not necessarily independent)with lifetimes 119879

119894 119894 = 1 2 119899 Suppose that the units

are working in a common operating environment whichis represented by a random vector Θ = (Θ

1 Θ2 Θ

119899)

independent of 1198791 1198792 119879

119899 and has an effect on the units

of the form

119883119894=119879119894

Θ119894

119894 = 1 2 119899 (40)

If Θ has support on (1infin)119899 then the components are

working in a harsh environment and if they have supporton (0 1)119899 then the components are working in a gentlerenvironment (see Ma [28]) In a harsh environment let 119879

119895isin

ASMRL for some 119895 = 1 2 119899ThenTheorem 19 states thatfor each 119885 with support on [0 1] we must have 119885119879

119895leMRL119879119895

Thus by taking 119885 = 1Θ119895 we must have 119879

119895Θ119895leMRL119879119895

Hence by (40) it stands that 119883119895leMRL119879119895 With a similar

discussion in a gentler environment if119883119895isinASMRL for some

119895 = 1 2 119899 then we must have 119879119895leMRL119883119895

In the following we state the preservation property of theASMRL class underweighted distribution Let119883 have densityfunction 119891 and survival function 119865 The following resultstates the preservation of the ASMRL class under weighteddistributions The proof is quite similar to that of Theorem 11and hence omitted

Theorem 29 Let 119861 be an increasing function and let119861(119904119909)119861(119904) increase in 119904 ge 0 for all 119909 isin (0 1] Then 119883 isASMRL implying that119883

119908is ASMRL

4 Conclusion

Due to economic consequences and safety issues it is nec-essary for the industry to perform systematic studies usingreliability concepts There exist plenty of scenarios where astatistical comparison of reliability measures is required inboth reliability engineering and biomedical fields In thispaper we have proposed a new stochastic order based onthe MRL function called proportional mean residual life(PMRL) order The relationships of this new stochastic orderwith other well-known stochastic orders are discussed Itwas shown that the PMRL order enjoys several reliabilityproperties which provide several applications in reliabilityand survival analysis We discussed several characterizationand preservation properties of this new order under somereliability operations To enhance the study we proposed anew class of life distributions called anti-star-shaped meanresidual life (ASMRL) class Several reliability properties ofthe new class as well as a number of applications in thecontext of reliability and survival analysis are included Ourresults provide new concepts and applications in reliabilitystatistics and risk theory Further properties and applicationsof the new stochastic order and the new proposed class canbe considered in the future of this research In particularthe following topics are interesting and still remain as openproblems

(i) closure properties of the PMRL order and theASMRLclass under convolution and coherent structures

8 Mathematical Problems in Engineering

(ii) discrete version of the PMRLorder and enhancing theobtained results related to the D-ASMRL class

(iii) testing exponentiality against the ASMRL class

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank two reviewers for theirvaluable comments and suggestions which were helpfulin improving the paper The authors would also like toextend their sincere appreciation to theDeanship of ScientificResearch at King Saud University for funding this ResearchGroup (no RG-1435-036)

References

[1] H M Ramos-Romero and M A Sordo-Dıaz ldquoThe propor-tional likelihood ratio order and applicationsrdquoQuestiio vol 25no 2 pp 211ndash223 2001

[2] F Belzunce J M Ruiz and M C Ruiz ldquoOn preservation ofsome shifted and proportional orders by systemsrdquo Statistics andProbability Letters vol 60 no 2 pp 141ndash154 2002

[3] C D Lai and M Xie Stochastic Ageing and Dependence forReliability Springer New York NY USA 2006

[4] R C Gupta and S N U Kirmani ldquoOn order relations betweenreliability measuresrdquo Communications in Statistics StochasticModels vol 3 no 1 pp 149ndash156 1987

[5] A A Alzaid ldquoMean residual life orderingrdquo Statistical Papersvol 29 no 1 pp 35ndash43 1988

[6] M Shaked and J G Shanthikumar Stochastic Orders SpringerSeries in Statistics Springer New York NY USA 2007

[7] A Muller and D Stoyan Comparison Methods for StochasticModels and Risks John Wiley New York NY USA 2002

[8] A K Nanda S Bhattacharjee and N Balakrishnan ldquoMeanresidual life function associated orderings and propertiesrdquoIEEE Transactions on Reliability vol 59 no 1 pp 55ndash65 2010

[9] M A Lariviere and E L Porteus ldquoSelling to a news vendoran analysis of price-only contractsrdquoManufacturing and ServiceOperations Management vol 3 no 4 pp 293ndash305 2001

[10] M A Lariviere ldquoA note on probability distributions withincreasing generalized failure ratesrdquo Operations Research vol54 no 3 pp 602ndash604 2006

[11] S Karlin Total Positivity Stanford University Press StanfordCalif USA 1968

[12] K JainH Singh and I Bagai ldquoRelations for reliabilitymeasuresof weighted distributionsrdquo Communications in StatisticsTheoryand Methods vol 18 no 12 pp 4393ndash4412 1990

[13] J Bartoszewicz and M Skolimowska ldquoPreservation of classesof life distributions and stochastic orders under weightingrdquoStatistics amp Probability Letters vol 76 no 6 pp 587ndash596 2006

[14] N Misra N Gupta and I D Dhariyal ldquoPreservation of someaging properties and stochastic orders by weighted distribu-tionsrdquo Communications in Statistics Theory and Methods vol37 no 5 pp 627ndash644 2008

[15] S Izadkhah andM Kayid ldquoReliability analysis of the harmonicmean inactivity time orderrdquo IEEE Transactions on Reliabilityvol 62 no 2 pp 329ndash337 2013

[16] M Kayid I A Ahmad S Izadkhah and A M AbouammohldquoFurther results involving the mean time to failure order andthe decreasing mean time to failure classrdquo IEEE Transactions onReliability vol 62 no 3 pp 670ndash678 2013

[17] S Izadkhah A H Rezaei Roknabadi and G R M BorzadaranldquoAspects of the mean residual life order for weighted distribu-tionsrdquo Statistics vol 48 no 4 pp 851ndash861 2014

[18] D Yue and J Cao ldquoSome results on the residual life at randomtimerdquo Acta Mathematicae Applicatae Sinica vol 16 no 4 pp435ndash443 2000

[19] X Li and M J Zuo ldquoStochastic comparison of residual life andinactivity time at a random timerdquo Stochastic Models vol 20 no2 pp 229ndash235 2004

[20] N Misra N Gupta and I D Dhariyal ldquoStochastic propertiesof residual life and inactivity time at a random timerdquo StochasticModels vol 24 no 1 pp 89ndash102 2008

[21] SMRoss Stochastic Processes JohnWileyNewYorkNYUSA1996

[22] R E Barlow and F Proschan StatisticalTheory of Reliability andLife Testing Silver Spring Md USA 1981

[23] X Ling P Zhao and P Li ldquoA note on the stochastic propertiesof a scale change random effects modelrdquo Statistics amp ProbabilityLetters vol 83 no 10 pp 2407ndash2414 2013

[24] E Fagiuoli and F Pellerey ldquoMean residual life and increasingconvex comparison of shock modelsrdquo Statistics amp ProbabilityLetters vol 20 no 5 pp 337ndash345 1994

[25] M Shaked and T Wong ldquoPreservation of stochastic orderingsunder random mapping by point processesrdquo Probability in theEngineering and Informational Sciences vol 9 no 4 pp 563ndash580 1995

[26] F Belzunce E Ortega and J M Ruiz ldquoThe Laplace order andordering of residual livesrdquo Statistics amp Probability Letters vol42 no 2 pp 145ndash156 1999

[27] M Kayid and S Izadkhah ldquoMean inactivity time functionassociated orderings and classes of life distributionsrdquo IEEETransactions on Reliability vol 63 no 2 pp 593ndash602 2014

[28] C Ma ldquoConvex orders for linear combinations of randomvariablesrdquo Journal of Statistical Planning and Inference vol 84no 1-2 pp 11ndash25 2000

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Mathematical Problems in Engineering

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