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1、RobustEstimationandTestingbyRobertG.StaudteandSimonJ.SheatherCopyright©1990JohnWiley&Sons,Inc.CHAPTERTWOEstimatingScale—FiniteSampleResultsInthischapterwewillreviewsometraditionalideasonscaleparameterestimation,andinparticularconsiderequivariantandunbiasedestimatorsofthee
2、xponentialscaleparameter.Ourgoalsaretoillustratetherobust-nessorlackthereofofsomeoftheclassicalestimators.Wewillintroducemixturemodelsforcontaminationandthefinitesamplebreakdownpointtoassistusinouranalysis,andfinallyconsidersomedifferentmethodsofestimatingthestandarderror
3、ofourestimators,includingthebootstrap.Thedifferencebetweenthelocationparameterandthescaleparameterestimationproblemsisthatintheformerashiftofthedistributionispos-sible,whileinthelatterafractionalormultiplicativechangeispossible.Mathematically,locationismodeledbyaddingthes
4、ameunknownconstanttoeachobservationfromafixeddistribution,whilescaleismodeledbymultiplyingeachobservationbyanunknownpositiveconstant.Thereadershouldbewarythatthesamplemean,median,andsomeotherestimatorscommonlyusedaslocationparameterestimatorsmayalsobereasonablescaleparame
5、terestimatorsinthescalecontext,atleastwhenmultipliedbysuitableconstants.ThelocationproblemistreatedinChapter4.2.1EXAMPLESTobeginwegiveexampleswhereascaleparameterisofinterest.Inthefirstexampleanexponentialmodelmaybeappropriate,andmanycharac-teristicsofthispopulationcanbee
6、stimated,startingwithanestimateofthescaleparameter.Inthesecondexampletheshapeofthedistributionisun-clear,butneverthelessanestimateofthescale(inthesenseofmagnitude)ofthepopulationisdesired.2122ESTIMATINGSCALE—FINITESAMPLERESULTS60-150-40-3020-100-1-T--r—-r—I501001502002503
7、00350rainfallintenthsofmillimetresFIGURE2.1Melbourne'sdailyrainfallonwinterraindaysfortheyears1981-1983.Example1:Melbourne'sDailyRainfall.RainfallvarieswiththeseasonsinMelbourne,Australia,soforthesakeoftimehomogeneitywerestrictattentiontothewintermonthsofJune,July,andAugu
8、st.Duringthisrainyseasonroughlyhalfthedayshavenomeasurablerainfall,andwewillhere-afterrestrictat
