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1、RobustEstimationandTestingbyRobertG.StaudteandSimonJ.SheatherCopyright?1990JohnWiley&Sons,Inc.CHAPTERFOURLocation-DispersionEstimationStatisticiansareofteninterestedinthe"locationproblem,"testingforapossibleshi?ofasymmetricdistribution,becauseitarisesnaturall

2、yinthedifferencesofmatchedpairsasexemplifiedinExample1ofSection4.1.2below.Wepostponesuchtestinguntilthenextchaptersincepowerful,ro-busttestsareusuallybasedonefficient,robustestimatorsoflocation.Inthischapterweconcentrateonsuchestimatorsforthelocationofasym-me

3、tricdistribution,butalsotreattheasymmetriccasewherelocationitselfneedstobedefined.Itisrareforthespread(dispersion)ofadistributiontobeknownwhenthelocationisunknown,soinpracticethereareatleasttwoquantitiesofinterest:locationanddispersion.Inthecaseofparametricmo

4、delsthelocation-scaleparameterrepresentstheunknownsoflocationanddisper-sion;numerousexamplesaregiveninSection4.2.Section4.3containsthemainresultsconcerningL-andM-estimatorsoflocationaswellasafewpropertiesoftheHodges-Lehmannestima-tor.InSection4.3.1thenotionof

5、locationisdefinedfornonparametricfamilieswhichmayincludeasymmetricdistributions.Afterdiscussionofanumberofexamplesoflinearcombinationsoforderstatistics,theclassofL-estimatorsforlocationisdefinedandtheinfluencefunctionderived.Weproveanimportantresult(duetoBick

6、elandLehmann)whichshowsthattheefficiencyoftrimmedmeansrelativetotheuntrimmedmeanisboundedbelowbyapositiveconstant.Thenwederivetheinfluencefunc-tionestimateofthestandarderrorofthetrimmedmean.Finallyweshowhowtoobtainanestimateofthestandarderrorofthemedianusingk

7、erneldensityestimators.Section4.3.3containsthemainresultsonM-estimators.FirstwefindtheinfluencefunctionandseethatitisproportionaltotheψfunctionwhichdefinestheM-equation(4.3.6);andwediscusshowtoexploitthis9596LOCATION-DISPERSIONΕβ?ΜΑΠΟΝfacttoobtainbothrobustne

8、ssandefficiencyofalocationestimator.Nextweshow(understrongconditions)thatM-estimatorsareconsistentandasymptoticallynormal,andwediscussHuber's(1964)asymptoticminimaxresults.InTheorem4.3wei