Nonparametric estimation for dependent data with a

Nonparametricestimationfordependentdatawithanapplicationtopaneltimeseries1JanJohannes2SuhasiniSubbaRao3February5,2008AbstractInthispaperweconsidernonparametricestimationfordependentdata,wheretheobservationsdonotnecessarilycomefromalinearprocess.Westudydensityestimationandalsodiscussassociatedproblemsinnonparametricregressionusingthe2-mixingdependencemeasure.Wecomparetheresultsunder2-mixingwiththosederivedundertheassumptionthattheprocessislinear.Inthecontextofpaneltimeserieswhereoneobservesdatafromseveralindividuals,itisoftentoostrongtoassumethejointlinearityofprocesses.Insteadthemethodsdevelopedinthispaperenableustoquantifythedependencethrough2-mixingwhichallowsfornonlinearity.Weproposeanestimatorofthepanelmeanfunctionandobtainitsrateofconvergence.Weshowthatundercertainconditionstherateofconvergencecanbeimprovedbyallowingthenumberofindividualsinthepaneltoincreasewithtime.Keywords:Densityestimation,nonparametricregression,2-mixing,nonlinearprocesses,paneltimeseries.AMS2000subjectclassications:Primary:62G05,62M10;Secondary:62G07,62G08.1IntroductionNonparametricestimationfordependentobservationshasalonghistoryinstatistics.Rosen-blatt[1970]rststudieddensityestimationfordependentdata.Sincethenseveralauthorshaveconsiderednonparametricestimationundervariousassumptions.Forexample,HallandHart[1990a],Giraitisetal.[1996],Mielniczuk[1997]andEstevasandVieu[2003]considerdensityestimationforlinearprocesseswhichhavelongmemory,whereasCheng1ThisworkwaspartiallysupportedbytheDFG(DA187/12-3).2UniversityHeidelberg,InstituteofAppliedMathematics,ImNeuenheimerFeld,294,D-69120Heidel-berg,Germany,[e-mail]3TexasA&MUniversity,DepartmentofStatistics,CollegeStation,Texas77843-3143,[e-mail]1arXiv:0706.3923v1[math.ST]27Jun2007andRobinson[1991]considerdensityestimationforrandomvariableswhicharenonlinearfunctionsofalinearprocess.Anotableresult,isthattheyshowiftheobservationswerefromalinearprocessandhaveshortmemory,thentheusualrateofconvergence,knownforindependentobservations,alsoholdsfordependentobservations.Ontheotherhand,forlongmemoryprocesses,therateofconvergenceisdierent.Interestingly,despitelongmemoryinuencingtherateofconvergence,thereisnoinuenceoflongmemoryonthebandwidthchoice,whichissameregardlessofshortorlongmemory.Inotherwords,iftheobservationscomefromalinearprocess,alargerbandwidthdoesnotimprovetherateofconvergenceofthedensityestimator.Similarresultscanalsobederivedfornonparametricregressionproblems(c.f.HallandHart[1990b],ChengandRobinson[1994]andCsorgoandMielniczuk[1995,1999,2001]).However,usuallyitisassumedthattheobservationscomefromalinearprocessorarefunctionsofalinearprocess.Inthecaseoflinearity,thejointdensityoftheobservationscanbecharacterised(insomesense)intermsoftheautocovariances.Itisthisrepresentationthatallowsforthemeansquarederrorofthenonparametricestimatortobederivedintermsoftheautocovariancefunction.Howeverthisresultdoesnotnecessarilyholdwhentheprocessisnonlinear.Theassumptionoflinearitycanberelaxedbyusingthenotionof2-mixing(seeBosq[1998]),andinthispaperweobtainratesofconvergenceforprocesseswhichare2-mixing.Unliketheautocovariancefunction,2-mixingcanbeconsideredasameasureofdependencebetweentworandomvariables(seeDenition3.1,below)andthe2-mixingsizequantiesthisdependence:alargemixingsizeindicateslittledependence,whereasasmallmixingsizeindicateslargedependence.The2-mixingsizecanbeestablishedforseveraltypesofpro-cesses,forexample,linearprocesses,seeAthreyaandPantula[1986],ClineandPu[1999],Chanda[1974]andtheAppendixA.4(notingthatstrongmixingimplies2-mixing,thoughtheconverseisnotnecessarilytrue)andnonlinearprocesses,seeMasryandTjstheim[1995],Bousamma[1998]andBasraketal.[2002].Assumingthatthe2-mixingsizeissucientlylarge,Bosq[1998]obtainstherateofconvergenceofseveralnonparametricesti-mators.Howeverdespite,therebeingextensiveliteratureonnonparametricestimationforlinearprocessesandsomeonnonparametricestimationforprocesseswhichare2-mixingwithasucientlylarge2-mixingsize,asfarasweareawareverylittleexistsonnonpara-metricestimationfornonlinearprocesseswhose2-mixingsizeisnotsucientlylarge(forexampletheARCH(1)process,whichisanonlinearprocessandcanhaveasmallmix-ingsize).Inthispaperweaddressthisissue,andconsidernonparametricestimationfordependentdataandformulatetheresultsintermsofthe2-mixingsize.Westudybothdensityestimationandalsononparametricregressionproblems.Anaturalapplicationofthemethodologyproposedinthispaperistopaneltimeseries,whereoneobservesseveralindividualsovertimeandassociatedwitheachindividualareregressorswhichareknowntoinuencetheindividual.Wenotethateveninthecasethatanindividualcomesfromalineartimeseries,thereisnoguaranteethatthedependencebetweenindividualsisalsolinear.Thereforewequantifythedependenceintermsofthe2-mixingsizewithinand2betweenindividualsovertime,andconsidernonparametricestimationforpaneltimeserieswithinthisframework.InSection3weconsiderkerneldensityestimation,inparticularweobtainthesamplingpropertiesoftheRosenblatt-Parzenkernelestimatorandobt