Robust estimates in generalized partially linear models

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1、TheAnnalsofStatistics2006,Vol.34,No.6,2856–2878DOI:10.1214/009053606000000858cInstituteofMathematicalStatistics,2006ROBUSTESTIMATESINGENERALIZEDPARTIALLYLINEARMODELS1ByGracielaBoente,XumingHeandJianhuiZhouUniversidaddeBuenosAiresandCONICET,UniversityofIllinoisatUrb

2、ana-ChampaignandUniversityofVirginiaInthispaper,weintroduceafamilyofrobustestimatesfortheparametricandnonparametriccomponentsunderageneralizedpar-tiallylinearmodel,wherethedataaremodeledbyyi

3、(xi,ti)∼F(·,µi)Twithµi=H(η(ti)+xiβ),forsomeknowndistributionfunctionFandli

4、nkfunctionH.Itisshownthattheestimatesofβareroot-nconsistentandasymptoticallynormal.ThroughaMonteCarlostudy,theperformanceoftheseestimatorsiscomparedwiththatoftheclas-sicalones.1.Introduction.Semiparametricmodelscontainbothaparametricandanonparametriccomponent.Somet

5、imes,thenonparametriccomponentplaystheroleofanuisanceparameter.Muchresearchhasbeendoneonestima-torsoftheparametriccomponentinageneralframework,aimingtoobtainasymptoticallyefficientestimators.Theaimofthispaperistoconsidersemi-parametricversionsofthegeneralizedlinearmo

6、delswheretheresponseyistobepredictedbycovariates(x,t),wherex∈Rpandt∈T⊂R.Itwillbeassumedthattheconditionaldistributionofy

7、(x,t)belongstothecanonicalexponentialfamilyexp[yθ(x,t)−B(θ(x,t))+C(y)]forknownfunctionsBandC.Thenµ(x,t)=E(y

8、(x,t))=B′(θ(x,t)),withB′denotingthed

9、eriva-tiveofB.Ingeneralizedlinearmodels[19],whichconstituteapopularap-proachformodelingawidevarietyofdata,itisoftenassumedthatthemeanismodeledlinearlythroughaknowninverselinkfunction,g,thatis,TarXiv:0708.0165v1[stat.ME]1Aug2007g(µ(x,t))=β0+xβ+αt.ReceivedJanuary2004

10、;revisedNovember2006.1SupportedinpartbyGrantsPICT03-00000-006277fromANPCYT,PID5505fromCONICET,X-094fromtheUniversidaddeBuenosAires,13900-6fromtheFundaci´onAntorchas,ArgentinaandU.S.NSFGrantDMS-01-02411.TheresearchbeganwhilethefirstauthorwasgrantedwithaGuggenheimfell

11、owship.AMS2000subjectclassifications.Primary62F35;secondary62G08.Keywordsandphrases.Kernelweights,partiallylinearmodels,rateofconvergence,robustes