ONPROPORTIONALREVERSEDHAZARDMODELFORDISCRETEDATASudheeshKKyandIshaDewanUniversityofHyderabadHyderabad46IndiaIndianStatisticalInstituteNewDelhi16IndiaEmailskkattuyahoocoinAbstract ID: 417318
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isid/ms/2009/11October16,2009http://www.isid.ac.in/estatmath/eprintsOnProportional(Reversed)HazardModelforDiscreteDataIshaDewanSudheeshK.K.IndianStatisticalInstitute,DelhiCentre7,SJSSMarg,NewDelhi{110016,India ONPROPORTIONAL(REVERSED)HAZARDMODELFORDISCRETEDATASudheeshK.K*yandIshaDewan***UniversityofHyderabad,Hyderabad-46,India.**IndianStatisticalInstitute,NewDelhi-16,India.E-mail:skkattu@yahoo.co.inAbstract.Inthispaper,weproposeaproportionalhazardsmodelfordiscretedataanalogoustotheversionforcontinuousdataandthenstudyitsproperties.Someageingpropertiesofthemodelarediscussed.Anewde nitionforreversedhazardrateisintroduced.Propertiesofproportionalreversedhazardsmodelarestudied.1.IntroductionTheliteratureonreliabilitytheorymainlydealswithnon-negativeabsolutelycontinuousran-domvariables.However,quietoftenwecomeacrosswithsituationswheretheproductlifecanbedescribedthroughnon-negativeintegervaluedrandomvariable.Forexample(i)adevicecanbemonitoredonlyoncepertimeperiodandtheobservationistakenasthenumberoftimeperiodssuccessfullycompletedpriortothefailureofadevice,(ii)apieceofequipmentmayoperateincyclesandwemeasurethenumberofcyclescompletedpriortofailure,forexamplenumberofcopiesmadebyaphotocopierbeforeitfails,(iii)numberofroadaccidentsinacityinagivenmonth.SeeChenandManatunga[7],Kemp[15]andYu[26]andthereferencestherein.Thereforeweneedtodeveloptools,analogoustothecontinuouscase,forstudyingthediscretefailuredata.LetXbediscreterandomvariablewithsupportN=f1;2;:::gorasubsetthereof.Supposep(x)=P(X=x),F(x)=P(Xx)=Pxj=1p(j)andR(x)=P(X�x)=P1j=x+1p(j)denote,respectively,theprobabilitymassfunction,distributionfunctionandreliability/survivalfunctionofX.ThenthehazardrateandthereversehazardratefunctionsofXde nedby1 2PROPORTIONAL(REVERSED)HAZARDMODELk(x)and(x),respectively,arek(x)=p(x) R(x 1);x=1;2;:::(1.1)and(x)=p(x) F(x);x=1;2;::::(1.2)Itisinterestingtonotethatinthediscretesetupboththehazardrateandthereversedhazardratecanbeinterpretedasaprobabilitywhichisnotthecaseinthecontinuouscase.IfXrepresentsthelifetimeofacomponentthenk(x)istheprobabilitythatthecomponentwillfailattimeX=xgiventhatithassurviveduptothetimebeforex.Similarly,(x)istheprobabilitythatthecomponentwillfailattimeX=x,givenitisknowntohavefailedbeforex.Becauseofthisniceinterpretationofk(x)and(x)asaprobabilityofaneventinthediscretecase,ithasreceivedwidespreadattention.Butmanyofthepropertiesofthehazardrateandthereversedhazardratewhichholdinthecontinuouscasedonotholdinthediscretecase.Xie,GoudoinandBracqmond[24]pointedoutseveralproblemsoranomaliesregardingthede nitionofk(x).Theyde nedanewhazardratefunction(seeRoyandGupta[22])indiscretesetupash(x)=lnR(x 1) R(x);x=1;2;::::(1.3)Xieetal.[24]alsodiscussedseveraladvantagesofthede nition(1.3).Andobtainedexactexpressionforhazardratefunctionh(x)forsomewellknowndiscretedistributions(seeLaiandXie[17],chapter6formoredetails).Itiseasytoverifythatthefunctionsk(x)andh(x)satisfytherelationsoftheformk(x)=1 exp( h(x));h(x)= ln(1 k(x));8x=1;2;::::(1.4)Hencebothk(x)andh(x)havethesamemonotonicityproperty.Thefunctionsk(x)andh(x)uniquelydeterminethedistributionofXthroughR(x)=xYj=1(1 k(j))(1.5)andR(x)=exp[ xXj=1h(j)];x=1;2;::::(1.6) PROPORTIONAL(REVERSED)HAZARDMODEL3Thesefunctionalformsareextensivelyusedformodellinglifetimedata.Xekalaki[23]pointedoutsituationswheretheproductlifeisdiscreteinnatureandgavecharacterizationresultsconcerningthegeometric,Waringandnegativehypergeometricdistri-butionsintermsofhazardrates.Gupta,GuptaandTripathi[11]characterizedcertainclassesofdiscretelifedistributionsbasedontheconvexitypropertiesofp(x).AdedicatedstudyondiscreteageingnotioncanbefoundinBracquemond,Goudoin,RoyandXie[3].Forare-centsurveyofdiscretereliabilityconceptsanddistributionsseeBracquemondandGaudoin[2].Morerecently,Kemp[15]didanexhaustivestudyontheageingbehaviourofdiscretelifedistributionsandgavesomenewinsightinthisdirection.Modellingdiscretelifetimedatausingtheproportionalhazardsmodel(PHM)orproportionalreversedhazardsmodel(PRHM)isrecentinterestinstatisticalliterature.SeeChenandManatunga[7],GuptaandGupta[10]andYu[26]andreferencesthereinformoredetails.ThisinspiredustostudythepropertiesofthePHM(PRHM)indiscretesetupanalogoustothoseinthecontinuouscase.Therestofthearticleisorganizedasfollows.InSection2wediscusscertainageingpropertiesofthediscretehazardrate.Insection3weproposeaproportionalhazardsmodelindiscretesetupandthenstudythepreservationofageingpropertiesundertheproposedmodel.InSection4weintroduceanewde nitionforreversedhazardrateforstudyingthepropertiesoftheproportionalreversedhazardsmodel.WeconcludeinSection5withindicationsofdirectionforfuturework.2.DiscreteAgeingNotionsClassi cationofclassesoflifetimedistributionsbasedonthenotionofageinghelpsinidenti -cationoftheunderlyingmodelwhichgeneratesthedata.Herewestudytheexistingnotionsofdiscreteageing.Wegivesomede nitionsofdiscretenotionofageingforcompleteness.De nition2.1.LetXbeadiscreterandomvariablede nedonN,then(i)Xissaidtobeincreasing(decreasing)inlikelihoodratio(ILR(DLR))iff(x)islogconcave(convex),ie.ifp(x+1)p(x 1)()p2(x)forallxinN.(ii)Xissaidtobeincreasing(decreasing)hazardrate(IHR(DHR))ifR(x)islogconcave(convex),ie.ifh(x)isincreasing(decreasing)inxforallxinN.(iii)Xissaidtobeincreasing(decreasing)hazardrateaverage(IHRA(DHRA))if[R(x)]1=x 4PROPORTIONAL(REVERSED)HAZARDMODELisdecreasing(increasing)inxforallxinN.(iv)Xissaidtobedecreasing(increasing)meanresiduallife(DMRL(IMRL))ifm(x)=E(X xjX�x)isdecreasing(increasing)inxforallxinN.(v)Xissaidtobenewbetter(worse)thanused(NBU(NWU))ifR(x+k)()R(x)R(k)forallx,kinN.(vi)Xissaidtobenewbetter(worse)thanusedinexpectation(NBUE(NWUE))ifm(x)()forallxinN,whereisthemeanofX.Remark2.1.Standardtextbooksgivetwode nitionsforIHRA(DHRA)andNBU(NWU)notionsindiscretesetup.Herewestatedauniquede nitioninviewofthenewhazardrateh(x)givenin(1.3).ForrecentdiscussionontheseaspectsseeLaiandXie[17].Nextwediscuss,thechainofimplicationsbetweentheseageingconcepts.Theorem2.1.TherandomvariableXhasILR(DLR)impliesithasIHR(DHR).Proof:SinceXhasILR,wehaver(x)=p(x+1) p(x)isdecreasinginx.Consider1 k(x+1) 1 k(x)=P1k=x+1p(k) p(x+1) P1k=xp(k) p(x)=[r(x+1) r(x)]+1Xk=1nkYt=1r(x+t)r(x+k+1) r(x)o:Whenr(x)&#x-465;&#x.237;isdecreasing(increasing)eachtermintheabovein nitesumislessthan(greaterthan)zero.Inviewoftherelation(1.4)theresultsfollows.Remark2.2.Thefollowingexampleillustratesthefactthattheconverseoftheabovetheoremisnottrue.Example2.1.ConsidertheprobabilitydistributionX12345p(x)0:100:10:400:300:10k(x)0:10:160:500:751r(x) 140:750:33Herek(x)isincreasing,butr(x)isnot,thatis,XhasIHRbutnotILR. PROPORTIONAL(REVERSED)HAZARDMODEL5Lemma2.1.TherandomvariableXisIHR(DHR)ifandonlyifR(x 1) R(x)isincreasing(de-creasing)forallx2N.Lemma2.2.AsucientconditionthattherandomvariableXhasDMRL(IMRL)isthatthesequencehs(x)iisdecreasing(increasing)forallx2N,wheres(x)=R(x)=R(x 1).Proof:Bysimplealgebrawecanshowthatthemeanresiduallifefunctionm(x)takestheformm(x)=1 R(x)1Xk=xR(k):Considerm(x+1) m(x)=1 R(x+1)1Xk=x+1R(k) 1 R(x)1Xk=xR(k)=[s(x+2) s(x+1)]+1Xk=2nkYt=2s(x+t)s(x+k+1) s(x+1)o:Whenhs(x)iisdecreasing(increasing)eachtermintheabovein nitesumislessthan(greaterthan)zero.Hencetheresult.Itiseasytoprovethefollowingtheorem.Theorem2.2.TherandomvariableXhasIHR(DHR)impliesithasDMRL(IMRL).ThefollowingtheoremisapartialconverseofTheorem3.2duetoMi[19].Theorem2.3.TherandomvariableXhasDMRL(IMRL)andif4m(x)=m(x+1) m(x)ispositive(negative),thenXhasIHR(DHR).ThefollowingtheoremisprovedinBracqmondetal.[3].Theorem2.4.TherandomvariableXhasIHR(DHR)impliesithasIHRA(DHRA).Theorem2.5.TherandomvariableXhasIHRA(DHRA)impliesithasNBU(NWU).Proof:SupposeXhasIHRA,thenbyde nition[R(x)]1 xisdecreasing,hencewehave[R(x+k)]1 x+k[R(x)]1 x: 6PROPORTIONAL(REVERSED)HAZARDMODELTakingxthpoweronbothsideswehave[R(x+k)]x x+kR(x)orR(x+k)R(x)[R(x+k)]k x+k:Againusingthefact[R(x+k)]1 x+k[R(k)]1 k,theaboveinequalitycanbewrittenasR(x+k)R(x)R(k):HenceXhasNBUandthereversingtheinequalitywehavetheprooffortheNWUcases.Theorem2.6.TherandomvariableXhasNBU(NWU)impliesithasNBUE(NWUE).FortheproofseeTheorem3.4ofKemp[15].Theorem2.7.TherandomvariableXhasDMRL(IMRL)impliesithasNBUE(NWUE).Proof:SupposeXhasDMRL(IMRL).ThenbyLemma2.2wehaveP1k=x 1R(k) R(x 1)()P1k=xR(k) R(x):SinceR(0)=1,theaboveinequalitycanbewrittenas1Xk=0R(k)()P1k=xR(k) R(x):Byde nition,=E(X)=P1k=1P(Xx)=P1k=0R(k),henceXhasNBUE(NWUE).Asinthecontinuouscasewehavethefollowingchainofimplicationsoftheabovementionedclasses.IHR)IHRA)NBU)NBUEandIHR)DMRL)NBUE:Nextwestudythestochasticorderingbetweentworandomvariables.De nition2.2.SupposethatX1;X2aretworandomvariablewithcorrespondingprobabilitymassfunctionsp1(x);p2(x)respectively,then PROPORTIONAL(REVERSED)HAZARDMODEL7(i)X2issaidtobesmallerthanX1inthelikelihoodratioorder(X1lrX2)ifp1(x)=p2(x)isincreasingforallxinN.(ii)X2issaidtobesmallerthanX1inthehazardrateorder(X1hrX2)ifR1(x)=R2(x)increasesforallxinN,whereR1(x),R2(x)arethereliabilityfunctionofX1,X2respectively.(iii)X2issaidtobesmallerthanX1inthereversedhazardrateorder(X1rhrX2)ifF1(x)=F2(x)increasesforallxinN,whereF1(x),F2(x)arethedistributionfunctionofX1,X2respectively.(iv)X2issaidtobesmallerthanX1inthestochasticorder(X1stX2)ifR1(x)R2(x)forallxinN.(v)X2issaidtobesmallerthanX1inthemeanresiduallifeorder(X1mrlX2)ifE(X1 kjX1�k)E(X2 kjX2�k)forallxinN.vi)X2issaidtobesmallerthanX1inthemeanresiduallifeorder(X1rmrlX2)ifE(k X1jX1k)E(k X2jX2k)forallxinN.Theorem2.8.X1lrX2impliesX1hrX2:Proof:SupposeX1lrX2.thenbyde nitionp1(x)=p2(x)isincreasinginx1.Hencep1(x+k)p2(x+k+1)p1(x+k+1)p2(x+k);8x;k2N:Addingp1(x+k)p2(x+k)onbothsidesandrearrangingtermswehave(p2(x+k)+p2(x+k+1))p1(x+k) p2(x+k)(p1(x+k)+p1(x+k+1)):Sincep1(x+k 1) p2(x+k 1)p1(x+k) p2(x+k)theaboveinequalitycanbewrittenas(p2(x+k)+p2(x+k+1))p1(x+k 2) p2(x+k 2)(p2(x+k)+p2(x+k+1)):Bysimilarargumentabove,wecanarriveat(p2(x)+p2(x+1)+:::+p2(x+k+1))p1(x) p2(x)(p1(x)+p1(x+1)+:::+p1(x+k+1)):Ask!1,theaboveinequalityreducestoR2(x)p1(x) p2(x)R1(x): 8PROPORTIONAL(REVERSED)HAZARDMODELorh1(x)h2(x):Thenexttheoremfollowsonsimilarlines.Theorem2.9.X1lrX2impliesX1rhrX2:Theorem2.10.X1hrX2impliesX1stX2:Proof:SupposeX1hrX2,thenbyde nition,R1(x)=R2(x)increasesforallxinN,ifandonlyR1(x 1) R2(x 1)R1(x) R2(x)orR1(x 1)R2(x)R1(x)R2(x 1):Puttingx=0;1;2;:::;weobtainR2(1)R1(1);R2(2)R2(1);:::HenceR2(x)R1(x);x=1;2;3;:::;whichimpliesX1stX2:Theorem2.11.X1rhrX2impliesX1stX2:TheproofissimilartothatoftheTheorem2.10.Theorem2.12.X1hrX2impliesX1mrlX2:Proof:SupposeX1hrX2,thenbyde nition,R1(x)=R2(x)increasesforallxinN,ifandonlyR1(x) R2(x)R1(x+1) R2(x+1)orR1(x+1) R1(x)R2(x+1) R2(x):Changingxtox+1andmultiplyingwithaboveinequalityandsimplifyingweobtainR1(x+2) R1(x)R2(x+2) R2(x): PROPORTIONAL(REVERSED)HAZARDMODEL9Proceedingasabovewehavethesequenceofinequalities,R1(x+k) R1(x)R2(x+k) R2(x);k=1;2;:::Hence1Xk=x+1R1(k) R1(x)1Xk=x+1R2(k) R2(x);whichisequivalenttoE(X1 kjX1�k)E(X2 kjX2�k):Theorem2.13.X1rhrX2impliesX1rmrlX2:SincewecanexpressreversedmeanresiduallifeasE(k X1jX1k)=1 F(x)x 1Xk=1F(k);theproofisevidentfromTheorem2.12.Hencewehavethefollowingchainofimplicationsinthediscretedomainwhichiswell-knowninthecontinuouscase.X1lrX2impliesX1hrX2impliesX1stX2X1hrX2impliesX1mrlX2:X1lrX2impliesX1rhrX2impliesX1stX2X1rhrX2impliesX1rmrlX2:3.ProportionalHazardsModelPredictionofthelifeofadieisverycrucialforthesuccessoftheforgingindustry.Thispredictionhelpstheforgingindustrytoestimatethecostofthedieandhencesupplythedietothecustomeratareasonablylowerprice.Reducedpricecanescalatethedemandfromthecustomer.Thusthestudyandanalysisoffailuretimeofthedieisimportantinthiscontext.Onecanalsoestimatethenumberofcomponentsthatcanbeproducedusingaparticulardie.Sincethelifetimeiscomputedasthenumberofmetalcomponentsproducedbythedie,discreteageingconceptsareofinterest.Theperformanceofthedieise ectedbythestressapplied, 10PROPORTIONAL(REVERSED)HAZARDMODELenvironmentalfactorsliketemperatureandpressure(seeChanandMeeker[4]).Proportionalhazardsmodelisapossiblemodelinthiscontext.Consideraseriessystemconsistingofnindependentcomponents.ThenthelifetimeofthesystemZ=min(X1;X2;:::;Xn)hasreliabilityfunctiongivenbyR(x)=[R(x)]n:(3.1)Inthecontinuouscasethemodel(3.1)isknownasthePHMwherethehazardratecorre-spondingtotherandomvariableZisproportionaltothehazardrateofindividualcomponents.However,themodel(3.1)doesnotyieldproportionalhazardsinthediscretesetupwhenweusek(x)asthehazardrateofX.ThisinspiredustoproposeanewPHMinthediscretesetup.Let G(y)=P(Y�y)bethereliabilityfunctionoftherandomvariableY.Nowwede netheproportionalhazardsmodelindiscretesetupas G(x)=[R(x)];�0:(3.2)Ifh(x)isthehazardrateofXthenthehazardratecorrespondingtoYisgivenbyh(x)=ln G(x 1) G(x)=ln[R(x 1)] [R(x)]=h(x):(3.3)ThehazardrateofYisproportionaltothatoftherandomvariableX.Hencethemodel(3.2)istheProportionalHazardsModelforthediscretedata.First,westudysomestructuralpropertiesofthemodel(3.2).ThereliabilityfunctionofYintermsofhazardrateisgivenby G(y)=exp yXk=1h(k)=exp yXk=1h(k):TheprobabilitymassfunctionofYcanbeexpressedasg(y)= G(y 1) G(y)=[R(y 1)] [R(y)]=[R(y 1)]1 exp( h(y)): PROPORTIONAL(REVERSED)HAZARDMODEL11TherthmomentofYisgivenbyE(Yr)=1Xk=1krg(k)=1Xk=1kr[R(k 1)] [R(k)]:Inparticular,E(Y)=1Xk=0 G(k)=1Xk=0[R(k)]E(Y2)=1Xk=0(2k 1) G(k)=1Xk=0(2k 1)[R(k)]:HencethevarianceisgivenbyV(Y)=1Xk=0(2k 1)[R(k)] 1Xk=0[R(k)]2:Inthefollowingtheorems,weprovethepreservationofageingpropertiesunderthetransfor-mation(3.2).Theorem3.1.TherandomvariableXhasIHR(DHR)ifandonlyifandYhasIHR(DHR).Theproofisimmediatefromthede nitionbynotingthath(x)=h(x).Theorem3.2.TherandomvariableXhasDMRL(IMRL)ifandonlyifYhasDMRL(IMRL).Proof:LetXhasDMRL.ToproveYisDMRLitisenoughtoshowthaths(y)iisdecreasingwheres(y)= G(y)= G(y 1).Thatiss(y+1)s(y).Supposes(y+1)s(y)or G(y+1) G(y) G(y) G(y 1):Using(2.1)weobtainhR(x+1) R(x)ihR(x) R(x 1)i; 12PROPORTIONAL(REVERSED)HAZARDMODELwhichgiveshR(x+1) R(x)ihR(x) R(x 1)i:ThenbyLemma2.2XisIMRL,whichisacontradiction.Henceonepartofthetheoremisproved.Bysimilarargumentwecanprovetheconverse.Theorem3.3.TherandomvariableXhasNBU(NWU)ifandonlyifYhasNBU(NWU).Proof:SupposethatXhasNBUproperty,thenbyde nitionforallx;k2NwehaveR(x)R(k)R(x+k):Takingthpoweronbothsideswehave[R(x)][R(k)][R(x+k)]or G(x) G(k) G(x+k):Nowbyde nitionYhasNBUproperty.Retracingtheabovestepsintheoppositedirectionwehavetheprooffortheconversepart.TheprooffortheNWUcaseissimilar.Asabovewecanprovethefollowingtheorem.Theorem3.4.TherandomvariableXhasIHRA(DHRA)ifandonlyifYhasIHRA(DHRA).Example3.1.ConsiderthediscreteParetodistributionwithreliabilityfunctionR(x)=d x+dc;c;d�0;x1:ThePHMde nedby(2.2)hasreliabilityfunction G(x)=d x+dc;c;d�0;x1:Thehazardratesh(x)andh(x)aregivenbyh(x)=ln(d=x 1+d)c (d=x+d)c=clnhx+d x 1+di PROPORTIONAL(REVERSED)HAZARDMODEL13andh(x)=ln(d=x 1+d)c (d=x+d)c=clnhx+d x 1+di:Clearlybothh(x)andh(x)aredecreasinginxwhichisconsistentwithcontinuousanalogue.TheresultsgiveninTheorem3.2,3.3and3.4canbeeasilyveri ed.Nextwestudythestochasticorderingoftworandomvariableunderthetransformation(3.2).SupposethatY1andY2aretworandomvariablerelatedwithX1andX2respectivelybythetransformation(3.2).Theorem3.5.IfX1lrX2thenY1lrY2.Proof:Letg1(x)andg2(x)betheprobabilitymassfunctionsofY1andY2respectively.Con-siderg2(x)=g1(x)=[R2(x 1)]1 exp( h1(x)).[R1(x 1)]1 exp( h2(x))=h[R2(x 1)]=[R1(x 1)]ih1 exp( h1(x)).1 exp( h2(x))iSupposeX1lrX2holds,thenbyde nitionp1(x)=p2(x)isincreasinginx.Thatisp2(x)=p1(x)isdecreasinginxsothatR2(x 1)=R1(x 1)isdecreasinginx.Henceh2(x)=h1(x)isdecreasinginxand[(1 exp( h1(x)))=(1 exp( h2(x)))]isdecreasinginx.Takingalltheseinaccounttherighthandsideofaboveequationisdecreasinginx.Hencebyde nitionY1lrY2.Theproofsofthefollowingtwotheoremsaresimpleandhenceomitted.Theorem3.6.X1hrX2ifandonlyifY1hrY2.Theorem3.7.X1stX2ifandonlyifY1stY2.4.ProportionalreversedhazardsmodelInlifetimedataanalysis,theconceptsofreversedhazardratehaspotentialapplicationwhenthetimeelapsedsincefailureisaquantityofinterestinordertopredicttheactualtimeoffailure.Thereversedhazardrateismoreusefulinestimatingreliabilityfunctionwhenthedataareleftcensoredorrighttruncated.Forthediscussiononreversedhazardrateincontinuous 14PROPORTIONAL(REVERSED)HAZARDMODELdomainseeBlock,SavitsandSingh[1],ChandraandRoy([5],[6]),Finkelstein[9],Gupta,GuptaandSankaran[13],KunduandGupta[16]andNandaandGupta[20]andthereferencetherein.Alittleworkhasbeencarriedoutindiscretesetupinconnectionwithreversedhazardrate.Recently,NandaandSengupta[21]havediscussedreversedhazardrateindiscretesetupandobtainedseveralinterestingresults.Gupta,NairandAsha[14]characterizedcertainclassofdiscretelifedistributionsbymeansofarelationshipbetweenreversedhazardrateandrighttruncatedexpectation.TheresultsbyGuptaetal.[14]enlightentheroleofreversedhazardrateandtherighttruncatedexpectationincharacterizingdiscretelifedistributionwhichwasnotcoveredbytheearlierauthors.AsparalleltocelebratedPHMmodelthePRHMhasbecomepopularinrecenttimes.ForsurveyandsomepropertiesofPRHMincontinuousdomainseeGuptaandGupta[10].Theproportionalreversedhazardsmodelindiscretesetuphaspotentialapplicationinsystemre-liability.Consideraparallelsystemwithnindependentcomponent.Ifthelifetimeofeachcomponenthasdistributionfunctionoftheform(discretelogistic)F(x)=1 e x=d 1+e (x c)=d:ThenthelifetimeofthesystemZ=max(X1;X2;:::;Xn)hasthedistributionfunctiongivenbyF(x)=h1 e x=d 1+e (x c)=din=[F(x)]n:(4.1)Incontinuousdomainthemodel(4.1)iscalledPRHM(seeGupta,GuptaandGupta[12]).AspointedoutinSection2themodel(4.1)dosenotfollowaproportionalreversedhazardsmodelindiscretesetupwhenweuse(x)asthereversedhazardrate.Henceweintroduceanewde nitionforthereversedhazardrateinlineofcontinuouscase.Besidesthemodel(4.1)ensuresproportionalreversedhazardsrates.De nition4.1.LetXbediscreterandomvariablewithsupportN.ThenewreversedhazardrateofXisde nedasr(x)=lnF(x) F(x 1):(4.2)Therationalebehindthisde nitionisasfollows.Inthecontinuouscase,thereversedhazard PROPORTIONAL(REVERSED)HAZARDMODEL15rateisde nedas(x)=F0(x) F(x)=dlnF(x) dx:Insteadoftaking[F(x) F(x 1)]forF0(x)whichleadstotheexpression(1.2),wecoulduse[lnF(x) lnF(x 1)]for[dlnF(x)=dx]sothat(4.2)follows.Notethatther(x)isnotboundedbyoneandisadditiveforparallelsystemasinthecontinuouscase.Thefunctionr(x)determinesthedistributionofXuniquelybytherelationF(x)=exp1Xk=x r(k):(4.3)HencethecumulativereversedhazardrateisgivenbyH(x)=1Xk=xr(k)= lnF(x):Similarresultholdswhenthelifetimesarecontinuousrandomvariables.Since(x)andr(x)arerelatedthrough(x)=1 e r(x);(4.4)both(x)andr(x)havesamemonotonicityproperties.Thatis(x)isincreasing/decreasingifandonlyifr(x)isincreasing/decreasinginx.NowconsiderthereversedhazardrateofZr(x)=lnR(x) R(x 1)=lnF(x) F(x 1)n=nlnF(x) F(x 1)=nr(x):Clearlyr(x)isparalleltor(x).Hencewede neaPRHMindiscretesetupasfollows.LetG(y)bethedistributionoftherandomvariableYandsupposethatG(x)isrelatedthroughF(x)asG(x)=[F(x)];�0:(4.5)Thenthemodel(4.5)iscalledtheProportionalReversedHazardsModelasthereversedhazardrateofYisproportionaltothatofX.TheprobabilitymassfunctionofYcanbeexpressedasg(y)=G(y) G(y 1)=[F(y)] [F(y 1)]=[F(y)]1 exp( r(y)): 16PROPORTIONAL(REVERSED)HAZARDMODELTherthmomentofYisgivenbyE(Yr)=1Xk=1krg(k)=1Xk=1kr[F(k)] [F(k 1)]:Inparticular,E(Y)=1Xk=0 G(k)=1Xk=0(1 [F(k)])E(Y2)=1Xk=0(2k 1) G(k)=1Xk=0(2k 1)(1 [F(k)]):HencethevarianceisgivenbyV(Y)=1Xk=0(2k 1)(1 [F(k)]) 1Xk=0(1 [F(k)])2:Nowweproveaninterestingresultsconcerningthecompoundingsum.De nition4.2.Arandomvariablede nedonNissaidtobedistributedasSibuya()ifp(x)satis es(seeDevroye[8])p(x)=( 1)x+1x;x=1;2;:::;01:Theorem4.1.ConsidertherandomvariableY=min(X1;X2;:::;XN),whereNisrandomandindependentofXi,i=1;2;:::N.IfthedistributionofNisSibuya(),thenYadmitaproportionalreversedhazardsmodel.Proof:Consider1 G(x)=P(min(X1;X2;:::;XN)&#x-3.2;≦x)=1Xk=1P(min(X1;X2;:::;XN)&#x-3.2;≦x)P(N=k)=1Xk=1[R(x)]k( 1)k+1k: PROPORTIONAL(REVERSED)HAZARDMODEL17OrG(x)=1+1Xk=1[R(x)]k( 1)k+1k=(1 [R(x)])=[F(x)]:Hencewehavethemodel(4.5).Nextweprovecertainagingpropertiesunderthetransformation(4.5).De nition4.3.LetXbediscreterandomvariablede nedonN(seeLiandXu[18]).(i)TherandomvariableXissaidtobeincreasing(decreasing)reversedhazardrate(IRHR(DRHR))ifF(x)islogconvex(concave),ie.ifr(x)isincreasing(decreasing)inx.(ii)TherandomvariableXissaidtobeincreasing(decreasing)reversedmeanresiduallife(IRMRL(DMRL))ifv(x)=E(x XjXx)isincreasing(decreasing)forallx.Thefollowingtheoremisimmediatefromthede nition.Theorem4.2.TherandomvariableXhasIRHR(DRHR)ifandonlyifYhasIRHR(DRHR).AsparalleltotheresultsofGuptaetal.[12]wehavethefollowingresult.Theorem4.3.TherandomvariableXhasIHR(DHR)and�1(1)thenYhasIHR(DHR).Theproofofthenexttheoremisgreatlyfacilitatedbythefollowinglemma.Lemma4.1.Let bearealfunctionde nedonanintervalNandletx1y1y2andx1x2y2(i)if isconvexonN,then (y1) (x1) y1 x1 (y2) (x2) y2 x2(ii)if isconcaveonN,thentheaboveinequalityisreversed.Theorem4.4.TherandomvariableXhasNBU(NWU)and�1(1)thenYhasNBU(NWU). 18PROPORTIONAL(REVERSED)HAZARDMODELProof:SupposethatXhasNBUthenforallx;t0wehaveR(x+t 1)R(x 1)R(t 1):AftersomesimplealgebrawecanshowthattheaboveinequalityissameasF(x+t 1) F(x 1)F(t 1) F(x 1)F(t 1):(4.6)Letx1=F(x 1)F(t 1),x2=F(x 1),y1=F(t 1)andy2=F(x+t 1).HencebytheLemma4.1,for (x)=[F(x)]and�1wehavetheinequality(F(x+t 1) F(x 1))([F(x 1)] [F(x 1)][F(t 1)])(F(x 1) F(x 1)F(t 1))([F(x+t 1)] [F(x 1)]):(4.7)Inviewoftheinequality(4.6),tosatisfy(4.7)onemusthave[F(x 1)] [F(x 1)][F(t 1)][F(x+t 1)] [F(x 1)]:Thiswillreducesto[R(x+t 1)][R(x 1)][R(t 1)];hencetheproofforNBUcase.Siceforthe1theinequalityintheLemma4.1canbereversedsothattheprooffortheNWUcasecanbeestablishedinsimilarline.Lemma4.2.AsucientconditionthattherandomvariableXhasIRMRListhatthesequencehu(x)iisincreasingforallx2N,whereu(x)=F(x)=F(x+1).Proof:Bysimplealgebrawecanshowthatv(x)takestheformv(x)=1 F(x)x 1Xk=1F(k): PROPORTIONAL(REVERSED)HAZARDMODEL19Considerv(x+1) v(x)=1 F(x+1)xXk=1F(k) 1 F(x)x 1Xk=1F(k)=F(1) F(x+1)+(u(x) u(x 1))+x 2Xk=1nkYt=1u(x t))(u(x) u(x (k+1)))o:Whenhu(x)iisincreasingeachtermintheabove 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