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1、AnIntroductiontoRobustEstimationwithRFunctionsRuggeroBellioDepartmentofStatistics,UniversityofUdineruggero.bellio@dss.uniud.itLauraVenturaDepartmentofStatistics,UniversityofPadovaventura@stat.unipd.itOctober2005Contents1Introduction22Estimationinscaleandl
2、ocationmodels72.1Thefunctionmeanforatrimmedmean....................82.2Thefunctionsmedianandmad.........................102.3Thehuberandhubersfunctions.......................122.4Anintroductiontotherlmfunction......................143Estimationinscaleandr
3、egressionmodels173.1M-estimators..................................183.2Theweightedlikelihood............................263.3Highbreakdownregression...........................293.4Bounded-influenceestimationinlinearregressionmodels..........324Bounded-influ
4、enceestimationinlogisticregression444.1Mallows-typeestimation............................454.2TheBiancoandYohaiestimator........................461IntroductionSince1960,manytheoreticaleffortshavebeendevotedtodevelopstatisticalproceduresthatareresistanttosm
5、alldeviationsfromtheassumptions,i.e.robustwithrespecttooutliersandstablewithrespecttosmalldeviationsfromtheassumedparametricmodel.Infact,itiswell-knownthatclassicaloptimumproceduresbehavequitepoorlyunderslightviolationsofthestrictmodelassumptions.Itisalso
6、well-knownthattoscreenthedata,toremoveoutliersandthentoapplyclassicalinferentialproceduresisnotasimpleandgoodwaytoproceed.Firstofall,inmultivariateorhighlystructureddata,itcanbedifficulttosingleoutoutliersoritcanbeevenimpossibletoidentifyinfluentialobservati
7、ons.Second,inplaceofrejectinganobservation,itcouldbebettertodown-weightuncertainobservations,althoughwemaywishtorejectcompletelywrongobservations.Moreover,rejectingoutliersreducesthesamplesize,couldaffectthedistributiontheory,andvariancescouldbeunderestima
8、tedfromthecleaneddata.Finally,empiricalevidenceshowsthatgoodrobustproceduresbehavequitebetterthantechniquesbasedontherejectionofoutliers.Robuststatisticalproceduresfocusinestimation,testinghypothesesandinregressionm
