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1、RobustEstimationandTestingbyRobertG.StaudteandSimonJ.SheatherCopyright©1990JohnWiley&Sons,Inc.CHAPTERTHREEEstimatingScale—AsymptoticResultsInstatistics,somepeopleworryaboutnotseeingtheforestforthetrees.Iliketolookatthebark.C.R.Blyth(c.1967)Thisisaratherlengthychaptersinceitnotonlycont
2、ainsthebasicideasonrobustnesswhichareusedinsubsequentchapters,butalsocontainssectionswhichflownaturallyonfromitintovariousareasofresearch.Thesemoreabstractsections,whicharestarred,maybeskippedatfirstreadingofthetextwithoutlossofcontinuity.Itseemsworthwhiletobeginthismaterialwitha.long
3、preamblesothatthestudentwillbegintohaveanoverviewandadirectedpaththroughit.Itmaybewisetoreturntothisintroductorymapifdirectionislost.InSection3.1,"Consistency,AsymptoticNormality,andEfficiency,"wereviewthestandardmaterialonconsistency,asymptoticnormality,andasymptoticrelativeefficienc
4、yofasequenceofestimators,whichweassumethestudenthassomeacquaintancewith.(Inthisconnection,wenotethatAppendixB.lcontainssomebasicconvergencetheoremswhichareusefulinstatistics,plusafewsimpleexercisesattheendtohelpthereaderre-viewthem.)Theseasymptoticpropertiesofestimatorsarepresentedher
5、einthelanguageofdescriptivemeasures,sothattheymaybeutilizedinei-therparametricornonparametricsettings.TheywillbeillustratedfortheestimatorsofscalediscussedinChapter2,wherefinitesampleresultswereobtained.Oneoftheobjectivesofthesectionistodemonstratethatinsomecontextsasymptoticefficienc
6、ycanbeausefulguidetothebehaviorofanestimatorevenforsamplesizesaslowas20.InSection3.2,"RobustnessConcepts,"weconsiderthreedifferentwaysofassessingtherobustnessofanestimatorT(F„)ofT(F),namely,thebreakdownpoint,influencefunction,andcontinuityofthedescriptivemea-4344ESTIMATINGSCALE—ASYMPT
7、OTICRESULTSsureTwhichdefinesit.Theseconceptsareoftendescribedasquantita-tiverobustness,infinitesimalrobustness,andqualitativerobustness.AllcanbederivedfromthedescriptivemeasureTwhichinducestheestimatorse-quencethroughT„=T(F„).TheseconceptswereintroducedbyHampel(1968)inhisPh.D.thesis.T
8、hebre
