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1、TheAnnalsofStatistics2009,Vol.37,No.5A,2445–2457DOI:10.1214/08-AOS645cInstituteofMathematicalStatistics,2009ASYMPTOTICEQUIVALENCEOFEMPIRICALLIKELIHOODANDBAYESIANMAPByMarianGrendar´1andGeorgeJudgeBelUniversityandUniversityofCalifornia,BerkeleyInthispaperweareinterestedinempiri
2、callikelihood(EL)asamethodofestimation,andweaddressthefollowingtwoproblems:(1)selectingamongvariousempiricaldiscrepanciesinanELframe-workand(2)demonstratingthatELhasawell-definedprobabilisticinterpretationthatwouldjustifyitsuseinaBayesiancontext.Usingthelargedeviationsapproach
3、,aBayesianlawoflargenumbersisdevelopedthatimpliesthatELandtheBayesianmaximumapos-terioriprobability(MAP)estimatorsareconsistentundermisspeci-ficationandthatELcanbeviewedasanasymptoticformofMAP.Estimatorsbasedonotherempiricaldiscrepanciesare,ingeneral,inconsistentundermisspecifi
4、cation.1.Introduction.Owen’sempiricallikelihood(EL)theorem([30]and[31])providesundertraditionalassumptionsabasisforformingconfidenceregionsformultivariatemeansandparametersinestimatingequations.ThebasicELideaistoproceedasifthesampleX1,X2,...,Xn,drawnfromanunknowndistributionr(
5、x)isi.i.d.andcanbemodeledasamultinomialdistributionbasedontheobservations.InferencefortheJunknownparam-etersisbasedonKestimatingequationsandanonparametriclikelihoodratiostatisticthatasymptoticallyhasachi-squaredistribution.Asaresult,ELisanattractiveorthodoxsemiparametricmetho
6、dofestimationandin-ferencewhosescopehasbeenextendedinseveralproductivedirections(e.g.,see[17]and[41]).BuildingonOwen’sELinsight,thispaperisconcernedwithusingem-arXiv:0908.3397v1[math.ST]24Aug2009piricallikelihoodasamethodofestimation(cf.[3,19,27,31]and[32],amongothers).Throug
7、haBayesianlawoflargenumbers(BLLN,TheoremReceivedFebruary2008;revisedJune2008.1SupportedbyVEGA1/3016/06Grant.AMS2000subjectclassifications.Primary62G05,62C10;secondary60F10.Keywordsandphrases.Maximumnonparametriclikelihood,estimatingequations,Bayesiannonparametricconsistency,Ba
8、yesianlargedeviations,L-divergence,P´olyasam-pling,rightcensoring,Ka
