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1、2342P.HallAbstractAbriefaccountisgivenofthemethodologyandtheoryforthebootstrap.Methodologyisdevelopedinthecontextoftheequationapproach,whichallowsattentiontobefocussedonspecificcriteriaforexcellence,suchascoverageerrorofaconfidenceintervalorexpectedvalueofabias-correctedestimator.Thi
2、sapproachutilizesadefinitionofthebootstrapinwhichthekeycomponentisreplacingatruedistributionfunctionbyitsempiricalestimator.OurtheoryisEdgeworthexpansionbased,andisaimedspecificallyatelucidatingpropertiesofdifferentmethodsforconstructingbootstrapconfidenceintervalsinavarietyofsetting
3、s.ThereaderinterestedinmoredetailthancanbeprovidedhereisreferredtotherecentmonographofHall(1992).1.IntroductionAbroadinterpretationofbootstrapmethodsarguesthattheyaredefinedbyreplacinganunknowndistributionfunction,F,byitsempiricalestimator,p,inafunctionalformforanunknownquantityofint
4、erest.Fromthisstandpoint,theindividualwhofirstsuggestedthatapopulationmean,p=xdF(x),scouldbeestimatedbythesamplemean,x=xdF(x),swasusingthebootstrap.Wetendtofavourthisdefinition,althoughweappreciatethattherearealternativeviews.Perhapsthemostcommonalternativeistoconferthenamebootstrapo
5、nproceduresthatuseMonteCarlomethodstoeffectanumericalapproximation.Whileweseethatthisdoeshaveitsmerits,wewouldargueagainstitontwogrounds.First,itissometimesconvenienttodrawadistinctionbetweentheessentiallystatisticalargumentthatleadstothesubstitutionorplug-inmethoddescribedintheprevi
6、ousparagraph,andtheessentiallynumericalargumentthatemploysaMonteCarloapproximationtocalculateafunctionalofF^.Theredoexiststatisticalprocedureswhichmarrythenumericalsimulationandstatisticalestimationintooneoperation,wherethesimulationisregardedasprimarilyastatisticalfeature.MonteCarlo
7、testingisonesuchprocedure;seeforexampleCh.39:MethodoioyyandTheoryfortheBootstrap2343Barnard(1963),Hope(I968)andMarriott(1979).OurdefinitionofthebootstrapwouldnotregardMonteCarlotestingasabootstrapprocedure.Thatmaybeseenaseitheranadvantageoradisadvantage,dependingononesview.Asecondobj
8、ectionthaton
