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1、RobustEstimationandTestingbyRobertG.StaudteandSimonJ.SheatherCopyright©1990JohnWiley&Sons,Inc.CHAPTERFIVETestingforaLocationParameterThepropertyofrobustnessIbelievetobeevenmoreimportantinpracticethanthatthetestshouldhavemaximumpowerandthatthestatisticsemployedshou
2、ldbefullyefficient.G.E.P.Box(1953)Wecontinuethelocationparametercontextofthelastchapter,butnowoureffortsaredirectedtotestsofhypothesesconcerningthelocationofasymmetricdistribution.Firstwegiveabriefhistoricalreviewofthet-testandalternativetests.Thenweformallydefine
3、theone-samplelocationproblem,thepowerfunctionfortestsofthenullhypothesisversusone-sidedalternatives,andtheasymptoticpowerfunctionforsequencesofal-ternativeswhichapproachthenullhypothesisattherightrate.Wethencalculatetheasymptoticpowerfunctionsforanumberoftests,nam
4、ely,thesigntest,thef-test,the2/î-trimmedr-test,andtheWilcoxontest.Theasymptoticpowerfunctionsofthesigntestandofthetrimmedr-testarecomparedwithfinitesamplepowerfunctionsandareseentobegoodapproximationsformoderatesamplesizes.AlongthewaywewilldefinethePitmanefficienc
5、yofonetesttoan-other,andexpressthisasymptoticrelativeefficiencyoftestsintermsoftheirefficacies.Theseasymptoticresultsareroundedoffbysomepracticalconsiderations,suchasfindingthesamplesizerequiredtoachievepoweragainstaspecificalternativeforafixedlevel.InSection5.3we
6、motivateamodelfor"undetectableoutliers,"whichallowsustomonitortheeffectsofoutliersontheteststhroughtheasymp-toticpowerfunction.Wewillseethatonlythesigntest,amongthosestud-ied,hasrobustnessoflevelforthiskindofcontamination.Therearegreatdifferencesamongtheothertests
7、,withtheStudent'sr-testfaringworst.Theasymptoticpowerfunctionisalsofoundtodependontheinfluencefunctionoftheestimatoroflocationcorrespondingtotheteststatistic.149150TESTINGFORALOCATIONPARAMETERFinally,inSection5.4wederivetheasymptoticpowerfunctionofthe/-testinthepr
8、esenceofautoregressivedependence,andfindthatpositivedependencecausesthetesttooverstatethesignificanceofthetest,whilesimultaneouslyloweringthepowerforlar
