Empirical likelihood based testing for regression

arXiv:0711.4218v2[math.ST]16Jul2008ElectronicJournalofStatisticsVol.2(2008)581–604ISSN:1935-7524DOI:10.1214/07-EJS152EmpiricallikelihoodbasedtestingforregressionIngridVanKeilegom∗Universit´ecatholiquedeLouvainInstituteofStatisticsVoieduRomanPays201348Louvain-la-NeuveBelgiume-mail:[e-mail]´esarS´anchezSellero†andWenceslaoGonz´alezManteiga†UniversidaddeSantiagodeCompostelaDepartamentodeEstat´ısticaeI.O.FacultaddeMatem´aticasCampusSur.15706SantiagodeCompostelaSpaine-mail:[e-mail];[e-mail]:Considerarandomvector(X,Y)andletm(x)=E(Y|X=x).WeareinterestedintestingH0:m∈MΘ,G={γ(·,θ,g):θ∈Θ,g∈G}forsomeknownfunctionγ,somecompactsetΘ⊂IRpandsomefunctionsetGofrealvaluedfunctions.Specificexamplesofthisgeneralhypothe-sisincludetestingforaparametricregressionmodel,ageneralizedlinearmodel,apartiallinearmodel,asingleindexmodel,butalsotheselectionofexplanatoryvariablescanbeconsideredasaspecialcaseofthishypothesis.Totestthisnullhypothesis,wemakeuseoftheso-calledmarkedem-piricalprocessintroducedby[4]andstudiedby[16]fortheparticularcaseofparametricregression,incombinationwiththemoderntechniqueofem-piricallikelihoodtheoryinordertoobtainapowerfultestingprocedure.Theasymptoticvalidityoftheproposedtestisestablished,anditsfinitesampleperformanceiscomparedwithotherexistingtestsbymeansofasimulationstudy.AMS2000subjectclassifications:Primary62E20;secondary62F03,62F05,62F40,62G08,62G10.Keywordsandphrases:Markedempiricalprocess,Modelcheckforre-gression,Nonlinearregression,Partiallinearmodel,Residuals.ReceivedNovember2007.∗FinancialsupportfromIAPresearchnetworksnr.P5/24andP6/03oftheBelgiangov-ernment(BelgianSciencePolicy)isgratefullyacknowledged.†FinancialsupportfromtheSpanishMinistryofScienceandTechnology(withadditionalEuropeanFEDERsupport)throughprojectMTM2005-00820.581I.VanKeilegometal./Empiricallikelihoodbasedtestingforregression582Contents1Introduction.................................5822Generaltestprocedure...........................5833Applicationofgeneraltesttospecificmodels..............5873.1Parametricmodels..........................5873.2Generalizedlinearmodels......................5893.3Selectionofvariables.........................5913.4Partiallinearmodels.........................5923.5Othermodels.............................5944Simulations.................................5944.1Parametricmodel...........................5944.2Generalizedlinearmodel.......................596Appendix:Proofs................................597References....................................6021.IntroductionAssumethatthedata(Xi,Yi)(i=1,...,n)areindependentreplicationsofarandomvector(X,Y),whereXisad-variatevectorandYisone-dimensional.Letm(x)=E(Y|X=x)forx∈IRd.WeareinterestedintestingH0:m∈MΘ,G={γ(·,θ,g):θ∈Θ,g∈G}(1.1)forsomeknownfunctionγ,somecompactsetΘ⊂IRpandsomefunctionsetGofrealvaluedfunctions.Specialcasesofthisgeneralnullhypothesisincludetestingforaparametricmodel(inwhichcasem(·)≡γ(·,θ)),ageneralizedlinearmodel(m(x)=γ(βtx,α)withθ=(α,β)),apartiallinearmodel(m(x)=ztθ+g(w)withx=(w,z)),butthetestprocedurecanalsobeusedfore.g.theselectionofexplanatoryvariables.Otherpossibilitiesnotincludedinthepaperaretestingfortheparametricformofthevariancefunction,thecomparisonofregressioncurves,etc.Theideaofthetestprocedurewedevelopinthispaperistomakeuseoftheso-calledmarkedempiricalprocessintroducedby[4]andstudiedby[16],combinedwiththemoderntechniqueofempiricallikelihoodtheoryinordertoobtainapowerfultestingprocedure.Alotofresearchongoodness-of-fittestsbasedonempiricalprocessideashasbeencarriedoutinthelasttenyears.Startingwiththealready-mentionedpa-perby[4]and[16],devotedtotestingparametricregressionmodels,thetheorywascontinuedwiththeproblemofcheckinggeneralizedlinearmodels(see[20]),andtheselectionofvariables(see[3]),etc.Seealso[25]fortesting,usingtheempiricalprocessideas,othermodelsofinterest:partiallinearmodels,reductionofthedimension,modelswithmultidimensionalresponseandheteroscedasticitytests.Forthecalibrationofthecriticalpointsassociatedtotheteststatistics,twoapproximationsweremainlyused:oneisbasedonthebootstrap(see[17])andtheotherisbasedonmartingaletransformations(see[19]).ForalternativeI.VanKeilegometal./Empiricallikelihoodbasedtestingforregression583testingproceduresbasedonsmoothingtechniques,see[23],andforrecentgen-eralizationsoftheempiricalregressionprocessideastodependentdata,seethepapersby[18],[6],[7]and[8].Theempiricallikelihood(seeforexamplethebookby[14])isatechniquedesignedtoconstructanonparametriclikelihoodforparametersofinterestinanonparametricorasemiparametricsettingwithnicepropertiestypicalfortheparametriclikelihood,asforexampletheWilks’theoremandtheBartlettcorrectionrecentlyprovedby[1].Therehasalsobeensomerecentinterestingoodness-of-fittestsbasedontheempiricallikelihoodintheregressioncontext.[9]proposeasieveempiricallikeli-hoodtestfortestinggeneralvarying-coefficientregressionmodels.[11]studythepropertiesoftheempiricallikelihoodinthepresenceofbothfiniteandinfinitedimensionalnuisanceparametersaswellaswhenthedatadimension