The Weibull-Geometric distribution

arXiv:0809.2703v1[stat.ME]16Sep2008TheWeibull-GeometricDistributionWagnerBarreto-Souzaa,AliceLemosdeMoraisaandGaussM.CordeirobaDepartamentodeEstat´ıstica,UniversidadeFederaldePernambuco,CidadeUniversit´aria,50740-540–Recife,PE,Brazil(e-mail:[e-mail],[e-mail])bDepartamentodeEstat´ısticaeInform´atica,UniversidadeFederalRuraldePernambuco,RuaDomManoeldeMedeiross/n,50171-900–Recife,PE,Brazil(e-mail:[e-mail])AbstractInthispaperweintroduce,forthefirsttime,theWeibull-Geometricdistributionwhichgeneralizestheexponential-geometricdistributionproposedbyAdamidisandLoukas(1998).Thehazardfunctionofthelastdistributionismonotonedecreasingbutthehazardfunctionofthenewdistributioncantakemoregeneralforms.UnliketheWeibulldistribution,theproposeddistributionisusefulformodelingunimodalfailurerates.Wederivethecumulativedistributionandhazardfunctions,theden-sityoftheorderstatisticsandcalculateexpressionsforitsmomentsandforthemomentsoftheorderstatistics.WegiveexpressionsfortheR´enyiandShannonentropies.Themaximumlikelihoodestimationprocedureisdiscussedandanalgo-rithmEM(Dempsteretal.,1977;McLachlanandKrishnan,1997)isprovidedforestimatingtheparameters.Weobtaintheinformationmatrixanddiscussinference.Applicationstorealdatasetsaregiventoshowtheflexibilityandpotentialityoftheproposeddistribution.keywords:EMalgorithm;Exponentialdistribution;Geometricdistribution;Hazardfunction;Informationmatrix;Maximumlikelihoodestimation;Weibulldis-tribution1IntroductionSeveraldistributionshavebeenproposedintheliteraturetomodellifetimedata.AdamidisandLoukas(1998)introducedthetwo-parameterexponential-geometric(EG)distribu-tionwithdecreasingfailurerate.Kus(2007)introducedtheexponential-Poissondistri-bution(followingthesameideaoftheEGdistribution)withdecreasingfailurerateand1discussedvariousofitsproperties.MarshallandOlkin(1997)presentedamethodforaddingaparametertoafamilyofdistributionswithapplicationtotheexponentialandWeibullfamilies.Adamidisetal.(2005)proposedtheextendedexponential-geometric(EEG)distributionwhichgeneralizestheEGdistributionanddiscussedvariousofitsstatisticalpropertiesalongwithitsreliabilityfeatures.ThehazardfunctionoftheEEGdistributioncanbemonotonedecreasing,increasingorconstant.TheWeibulldistributionisoneofthemostcommonlyusedlifetimedistributioninmodelinglifetimedata.Inpractice,ithasbeenshowntobeveryflexibleinmodelingvarioustypesoflifetimedistributionswithmonotonefailureratesbutitisnotusefulformodelingthebathtubshapedandtheunimodalfailurerateswhicharecommoninreliabilityandbiologicalstudies.InthispaperweintroduceaWeibull-geometric(WG)distributionwhichgeneralizestheEGandWeibulldistributionsandstudysomeofitsproperties.Thepaperisorganizedasfollows.InSection2,wedefinetheWGdistributionandplotitsprobabilitydensityfunction(pdf).InSection3,wegivesomepropertiesofthenewdistribution.Weobtainthecumulativedistributionfunction(cdf),survivorandhazardfunctionsandthepdfoftheorderstatistics.Wealsogiveexpressionsforitsmomentsandforthemomentsoftheorderstatistics.TheestimationbymaximumlikelihoodusingthealgorithmEMisstudiedinSection4andinferenceisdiscussedinSection5.IllustrativeexamplesbasedonrealdataaregiveninSection6.Finally,Section7concludesthepaper.2TheWGdistributionTheEGdistribution(AdamidisandLoukas,1998)canbeobtainedbycompoundinganexponentialwithageometricdistribution.Infact,ifXfollowsanexponentialdistributionwithparameterβZ,whereZisageometricvariablewithparameterp,thenXhastheEGdistributionwithparameters(β,p).SincetheWeibulldistributiongeneralizestheexponentialdistribution,itisnaturaltoextendtheEGdistributionbyreplacingintheabovecompoundingmechanismtheexponentialbytheWeibulldistribution.Supposethat{Yi}Zi=1areindependentandidenticallydistributed(iid)randomvari-ablesfollowingtheWeibulldistributionW(β,α)withscaleparameterβ0,shapeparameterα0andpdfg(x;β,α)=αβαxα−1e−(βx)α,x0,andNadiscreterandomvariablehavingageometricdistributionwithprobabilityfunc-tionP(n;p)=(1−p)pn−1forn∈Nandp∈(0,1).LetX=min{Yi}Ni=1.ThemarginalpdfofXisf(x;p,β,α)=αβα(1−p)xα−1e−(βx)α{1−pe−(βx)α}−2,x0,(1)whichdefinestheWGdistribution.Itisevidentthat(1)ismuchmoreflexiblethantheWeibulldistribution.TheEGdistributionisaspecialcaseoftheWGdistributionforα=1.Whenpapproacheszero,theWGdistributionleadstotheWeibullW(β,α)distribution.Figure1plotstheWGdensityforsomevaluesofthevectorφ=(β,α)whenp=0.01,0.2,0.5,0.9.Forallvaluesofparameters,thedensitytendstozeroasx→∞.20.00.51.01.52.02.53.00.00.51.01.52.0ϕ=(0.5,0.8)xf(x)p=0.01p=0.2p=0.5p=0.90.00.51.01.52.02.53.00.00.51.01.52.0ϕ=(0.5,1.5)xf(x)p=0.01p=0.2p=0.5p=0.90.00.51.01.52.02.53.00.00.51.01.52.0ϕ=(0.9,2)xf(x)p=0.01p=0.2p=0.5p=0.90.00.51.01.52.02.53.00.00.51.01.52.0ϕ=(0.9,5)xf(x)p=0.01p=0.2p=0.5p=0.9Figure1:PdfoftheWGdistributionforselectedvaluesoftheparameters.Forα1,theWGdensityisunimodal(seeappendixA)andthemodex0=β−1u1/α0isobtainedbysolvingthenonlinearequationu0+pe−u0u0+α−1α=α−1α.(2)ThepdfoftheWGdistributioncanbeexpressedasaninfinitemixtureofWeibulldis-tributionswiththesameshapeparameterα.If|z|1andk0,wehavetheseriesr