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1、Chapter5EstimationinNonparametricModelsEstimationmethodsstudiedinthischapterareusefulfornonparametricmodelsaswellasforparametricmodelsinwhichtheparametricmodelassumptionsmightbeviolated(sothatrobustestimatorsarerequired)orthenumberofunknownparametersisexceptionallylar
2、ge.SomesuchmethodshavebeenintroducedinChapter3;forexample,themethodsthatproduceUMVUE’sinnonparametricmodels,theU-andV-statistics,theLSE’sandBLUE’s,theHorvitz-Thompsonestimators,andthesample(central)moments.Thetheoreticaljustificationforestimatorsinnonparametricmodels,h
3、owever,reliesmoreonasymptoticsthanthatinparametricmodels.Thismeansthatapplicationsofnonparametricmethodsusuallyrequirelargesamplesizes.Also,estimatorsderivedusingparametricmethodsareasymp-toticallymoreefficientthanthosebasedonnonparametricmethodswhentheparametricmodelsa
4、recorrect.Thus,tochoosebetweenaparametricmethodandanonparametricmethod,weneedtobalancetheadvantageofrequiringweakermodelassumptions(robustness)againstthedrawbackoflosingefficiency,whichresultsinrequiringalargersamplesize.ItisassumedinthischapterthatasampleX=(X1,...,Xn)i
5、sfromapopulationinanonparametricfamily,whereXi’sarerandomvectors.5.1DistributionEstimatorsInmanyapplicationsthec.d.f.’sofXi’saredeterminedbyasinglec.d.f.FonRd;forexample,X’sarei.i.d.randomd-vectors.Inthissection,wei3193205.EstimationinNonparametricModelsconsidertheest
6、imationofForF(t)forseveralt’s,underanonparametricmodelinwhichverylittleisassumedaboutF.5.1.1Empiricalc.d.f.’sini.i.d.casesFori.i.d.randomvariablesX1,...,Xn,theempiricalc.d.f.Fnisdefinedin(2.28).Thedefinitionoftheempiricalc.d.f.basedonX=(X1,...,Xn)inthecaseofX∈Rdisanalog
7、ouslygivenbyiXn1dFn(t)=I(−∞,t](Xi),t∈R,(5.1)ni=1where(−∞,a]denotestheset(−∞,a1]×···×(−∞,ad]foranya=(a,...,a)∈Rd.Similartothecaseofd=1(Example2.26),F(t)as1dnanestimatorofF(t)hasthefollowingproperties.Foranyt∈Rd,nF(t)nhasthebinomialdistributionBi(F(t),n);Fn(t)isunbiased
8、withvarianceF(t)[1−F(t)]/n;Fn(t)istheUMVUEundersomenonparametricmod-√els;andFn(t)isn-consistentforF(t).Foranymfixeddistinctpo
