2008 Weighted empirical likelihood in some two-sample semiparametric models with various types of censored data

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1、TheAnnalsofStatistics2008,Vol.36,No.1,147166DOI:10.1214/009053607000000695©InstituteofMathematicalStatistics,2008WEIGHTEDEMPIRICALLIKELIHOODINSOMETWO-SAMPLESEMIPARAMETRICMODELSWITHVARIOUSTYPESOFCENSOREDDATABYJIAN-JIANREN1UniversityofCentralFloridaInthisarticle,thew

2、eightedempiricallikelihoodisappliedtoageneralsettingoftwo-samplesemiparametricmodels,whichincludesbiasedsam-plingmodelsandcase-controllogisticregressionmodelsasspecialcases.Forvarioustypesofcensoreddata,suchasrightcensoreddata,doublycensoreddata,intervalcensoreddat

3、aandpartlyinterval-censoreddata,theweightedempiricallikelihood-basedsemiparametricmaximumlikelihoodestimator(θ˜n,F˜n)fortheunderlyingparameterθ0anddistributionF0isderived,andthestrongconsistencyof(θ˜n,F˜n)andtheasymptoticnormalityofθ˜nareestablished.Underbiasedsamp

4、lingmodels,theweightedempiricallog-likelihoodratioisshowntohaveanasymptoticscaledchi-squareddistribu-tionforcensoreddataaforementioned.Forrightcensoreddata,doublycen-√soreddataandpartlyinterval-censoreddata,itisshownthatn(F˜n−F0)weaklyconvergestoacenteredGaussianpr

5、ocess,whichleadstoaconsistentgoodness-of-fittestforthecase-controllogisticregressionmodels.1.Introduction.Considerthefollowingtwo-samplesemiparametricmodel:X1,...,Xnisarandomsamplewithdensityf0(x),0(1.1)Y1,...,Ynisarandomsamplewithdensityg0(x)=ϕ(x;θ0)f0(x),1wherethe

6、twosamplesareindependent,andϕ(x;θ0)isaknownfunctionwithx∈Randauniqueunknownparameterθ0∈Rq,whilef0andg0arethedensityfunctionsofunknownnonnegativedistributionfunctions(d.f.)F0andG0,respec-tively.Thismodel(1.1)includesbiasedsamplingmodels(Vardi[32])andcase-controllogi

7、sticregressionmodels(PrenticeandPyke[22])asspecialcases,forwhichtherehasnotbeenanypublishedworkdealingwithcensoreddata.Inthisarticle,westudymodel(1.1)whenatleastoneofthetwosamplesisnotcompletelyobservableduetocensoring.Asfollows,weuserandomsampleX1,...,Xntoil-0lust

8、ratethecensoringmodelsunderconsiderationhere,whileExamples1and2discussbiasedsamplingmodelsandcase-controllogisticregressionmodels,re-spectively.R