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1、Biometrika(1980),67,1,pp.59-6359PrintedinOreatBritainExactsaddlepointapproximationsBYH.E.DANIELSStatisticalLaboratory,UniversityofCambridgeSUMMARYTherenormalizedsaddlepointapproximationtotheprobabilitydensityofanestimatoroftenhasasurprisinglylowrelativeerroroverthe
2、wholeadmissiblerangeoftheparameter.Inparticularitisknowntobeexactforcertaindensities.Thisraisesthequestionofhowtocharacterizetheclassofsuchexactcases.ThedensityofthemeanofaunivariaterandomDownloadedfromsampleisdiscussed.Itisshownthatthenormal,gammaandinversenormala
3、retheonlypossibledensitiesforwhichtherenormalizedsaddlepointapproximationreproducesexactlythedensityofthemean.Somekeywords:Asymptoticexpansion;Sumofindependentrandomvariables.http://biomet.oxfordjournals.org/1.INTRODUCTIONTherehasrecentlybeenarevivalofinterestinsad
4、dlepointapproximationstoprobabilitydensitiesandtailareasofthedistributionofestimators;see,forexample,Hampel(1973),Phillips(1978),Barndorff-Nielsen&Cox(1979)andanasyetunpublishedpaperbyJ.Durbin.Suchanapproximation,renormalizedifnecessary,istypicallyaccurateforquites
5、mallsamplesizesn,andinmanycasestherelativeerrorisuniformlyOfji'1)overthewholeatHarvardUniversityonOctober11,2012admissiblerangeoftheparameter.Incertaincasestherenormalizedsaddlepointapproxi-mationevenreproducesthetruedensityexactly.Forexample,inthecaseofthenormal,g
6、ammaandinversenormaldensitiestherenormalizedsaddlepointapproximationtothedensityofthemeanofarandomsampleofsizenturnsouttobeexactforalln.Thereareotherexamplesconcerningtheratioofmeans,andconditionalestimators(Daniels,1954;Barndorff-Nielsen&Cox,1979).Oneisthereforele
7、dtoenquireabouttheclassofdensitieshavingthisexactnessproperty.Inthepresentpaperweinvestigatethesimplestsituation,namelythatofthemeanofaunivariaterandomsample,discussedin§2-4ofBarndorff-Nielsen&Cox(1979).Weshowthatthenormal,gammaandinversenormalaretheonlypossibleden
8、sitiesforwhichtherenormalizedsaddlepointapproximationisexact.2.THESADDLEPOINTEXPANSIONThenotationofDaniels(1954)isused.Letf(x)betheprobabilityden
