Explicit expressions for moments of the beta Weibu

arXiv:0809.1860v1[stat.ME]10Sep2008ExplicitexpressionsformomentsofthebetaWeibulldistributionGaussM.Cordeiroa,∗,AlexandreB.Simasb,†andBorkoD.Stoˇsi´ca,‡aDepartamentodeEstat´ısticaeInform´atica,UniversidadeFederalRuraldePernambuco,RuaDomManoeldeMedeiross/n,DoisIrm˜aos,52171-900Recife-PE,BrasilbAssocia¸c˜aoInstitutoNacionaldeMatem´aticaPuraeAplicada,IMPA,EstradaD.Castorina,110,Jd.Botˆanico,22460-320,RiodeJaneiro-RJ,BrasilAbstractThebetaWeibulldistributionwasintroducedbyFamoyeetal.(2005)andstudiedbytheseauthors.However,theydonotgiveexplicitexpressionsforthemoments.Wenowderiveexplicitclosedformexpressionsforthecumulativedistri-butionfunctionandforthemomentsofthisdistribution.Wealsogiveanasymp-toticexpansionforthemomentgeneratingfunction.Further,wediscussmaximumlikelihoodestimationandprovideformulaefortheelementsoftheFisherinforma-tionmatrix.Wealsodemonstratetheusefulnessofthisdistributiononarealdataset.Keywords:BetaWeibulldistribution,Fisherinformationmatrix,Maximumlikeli-hood,Moment,Weibulldistribution.1IntroductionTheWeibulldistributionisapopulardistributionwidelyusedforanalyzinglifetimedata.WeworkwiththebetaWeibull(BW)distributionbecauseofthewideapplicabilityoftheWeibulldistributionandthefactthatitextendssomerecentdevelopeddistributions.Thisgeneralizationmayattractwiderapplicationinreliabilityandbiology.WederiveexplicitclosedformexpressionsforthedistributionfunctionandforthemomentsoftheBWdistribution.Anapplicationisillustratedtoarealdatasetwiththehopethatitwillattractmoreapplicationsinreliabilityandbiology,aswellasinotherareasofresearch.TheBWdistributionstemsfromthefollowinggeneralclass:ifGdenotesthecu-mulativedistributionfunction(cdf)ofarandomvariablethenageneralizedclassofdistributionscanbedefinedbyF(x)=IG(x)(a,b)(1)∗Correspondingauthor.E-mail:[e-mail]†E-mail:[e-mail]‡E-mail:[e-mail]1fora0andb0,whereIy(a,b)=By(a,b)B(a,b)=Ry0wa−1(1−w)b−1dwB(a,b)istheincompletebetafunctionratio,By(a,b)istheincompletebetafunction,B(a,b)=Γ(a)Γ(b)/Γ(a+b)isthebetafunctionandΓ(.)isthegammafunction.Thisclassofgeneralizeddistributionshasbeenreceivingincreasedattentionoverthelastyears,inparticularaftertherecentworksofEugeneetal.(2002)andJones(2004).Eugeneetal.(2002)introducedwhatisknownasthebetanormaldistributionbytakingG(x)in(1)tobethecdfofthenormaldistributionwithparametersμandσ.TheonlypropertiesofthebetanormaldistributionknownaresomefirstmomentsderivedbyEugeneetal.(2002)andsomemoregeneralmomentexpressionsderivedbyGuptaandNadarajah(2004).Morerecently,NadarajahandKotz(2004)wereabletoprovideclosedformexpressionsforthemoments,theasymptoticdistributionoftheextremeorderstatisticsandtheestimationprocedureforthebetaGumbeldistribution.Anotherdistributionthathappenstobelongto(1)isthelogF(orbetalogistic)distribution,whichhasbeenaroundforover20years(Brownetal.,2002),evenifitdidnotoriginatedirectlyfrom(1).Whilethetransformation(1)isnotanalyticallytractableinthegeneralcase,theformulaerelatedwiththeBWturnoutmanageable(asitisshownintherestofthispaper),andwiththeuseofmoderncomputerresourceswithanalyticandnumericalcapabilities,mayturnintoadequatetoolscomprisingthearsenalofappliedstatisticians.ThecurrentworkrepresentsanadvanceinthedirectiontracedbyNadarajahandKotz(2006),contrarytotheirbeliefthatsomemathematicalpropertiesoftheBWdistributionarenottractable.Thus,following(1)andreplacingG(x)bythecdfofaWeibulldistributionwithparameterscandλ,weobtainthecdfoftheBWdistributionF(x)=I1−exp{−(λx)c}(a,b)(2)forx0,a0,b0,c0andλ0.Thecorrespondingprobabilitydensityfunction(pdf)andthehazardratefunctionassociatedwith(2)are:f(x)=cλcB(a,b)xc−1exp{−b(λx)c}[1−exp{−(λx)c}]a−1,(3)andτ(x)=cλcxc−1exp{−b(λx)c}[1−exp{−(λx)c}]a−1B1−exp{−(λx)c}(a,b),(4)respectively.Simulationfrom(3)iseasy:ifBisarandomnumberfollowingabetadistributionwithparametersaandbthenX={−log(1−B)}1/c/λwillfollowaBWdistributionwithparametersa,b,candλ.SomemathematicalpropertiesoftheBWdistributionaregivenbyFamoyeetal.(2005)andLeeetal.(2006).Graphicalrepresentationofequations(3)and(4)forsomechoicesofparametersaandb,forfixedc=3andλ=1,aregiveninFigures1and2,respectively.ItshouldbenotedthatasingleWeibulldistributionfortheparticularchoiceoftheparametersc2andλisheregeneralizedbyafamilyofcurveswithavarietyofshapes,showninthesefigures.Therestofthepaperisorganizedasfollows.InSection2,weobtainsomeexpansionsforthecdfoftheBWdistribution,andpointoutsomespecialcasesthathavebeenconsideredintheliterature.InSection3,wederiveexplicitclosedformexpressionsforthemomentsandpresentskewnessandkurtosisfordifferentparametervalues.Section4givesanexpansionforitsmomentgeneratingfunction.InSection5,wediscussthemaximumlikelihoodestimationandprovidetheelementsoftheFisherinformationmatrix.InSection6,anapplicationtorealdataispresented,andfinally,inSection7,weprovidesomeconclusions.Intheappendix,twoidentitiesneededinSection3arederived.0,00,51,01,52,02,53,00,00,20,40,60,81,01,21,41,61,82,0a=0.5,b=0.5a=2.0,b=2.0a=0.5,b=2.0a=2.0,b=0.5Weibullf(x)xFigure1:Theprobabilitydensityfunction(3)oftheBWdistribution,forseveralvalue