20204060801000608112nPSfragreplacements7enTnTn7enFig1Thisplotshowsthatandapproachand respectivelyasnincreasesTosummarizehE8hErTnTn7en7 JBCrjforlarg ID: 337765
Download Pdf The PPT/PDF document "Standard Errors of Mean Variance and Sta..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
1StandardErrorsofMean,Variance,andStandardDeviationEstimatorsSangtaeAhnandJeffreyA.FesslerEECSDepartmentTheUniversityofMichiganJuly24,2003I.INTRODUCTIONWeoftenestimatethemean,variance,orstandarddeviationfromasampleof elementsandpresenttheestimateswithstandarderrorsorerrorbars(inplots)aswell.Astandarderrorofastatistic(orestimator)isthe(estimated)standarddeviationofthestatistic.Anerrorbaris,inaplot,alinewhichiscenteredattheestimatewithlengththatisdoublethestandarderror.Standarderrorsmeanthestatisticaluctuationofestimators,andtheyareimportantparticularlywhenonecomparestwoestimates(forexample,whetheronequantityishigherthantheotherinastatisticallymeaningfulway).Inthisnotewereviewthestandarderrorsoffrequentlyusedestimatorsofthemean,variance,andstandarddeviation.II.NORMALONESAMPLEPROBLEMLet \nbearandomsamplefrom \r wherebothandareunknownparameters.Dene,forconve-nience,twostatistics(samplemeanandsamplevariance): \n and \n \r !A.MeanEstimatorTheuniformlyminimumvarianceunbiased(UMVU)es-timatorofis[1,p.92].Since#" \r$%&' ,thestandarderrorofis)(*,+-/.10325467 Hence86(*89'7 .For8,seeSubsectionII-C.B.VarianceEstimatorNotefrom[1,p.92]that isUMVUfor&andthat\r ";:\n=(1)Sincethechi-squareddistributionwith&#x&00; degreesoffreedom?:9\n=hasavarianceofBC\r D [1,p.31],thestandarderrorof isFEHGIKJ-/.10AL NMOQPB R Hence8SEG JB'\r .ItisusefultonoteSEQG'8FEHG3' PB Since&and havethesquareoftheunitsof ,oftenitispreferabletoreportestimatesof,asdescribednext.C.StandardDeviationEstimatorTheUMVUestimatorofisT\n [1,p.92]whereT\nUP BV\r\n=V\r\nP BW3XYFZC[ \]H^G`_XYSZa[\G1_wherethesecondformismorenumericallystableforlargevaluesof whenusingthelngammafunction.BysettingT\nb, isacommonchoiceinpracticebutitisslightlybiased.Since7 "O:\n=[see(1)]andthechidistributionwith c degreesoffreedom\r:d\n=hasvariance[2,p.49:typocorrected]eF\n BUf R B V\r\nV\r\n=NgthestandarderrorofT\n isFh\EJ-/.10ALT\n M;T\nPei\n Toinvestigatetheasymptoticbehaviorofh\E,weneedthefollowingapproximation[3,P.602]:V\r\nV\r\n=P j Bf k\r j Ilnmf gog(2)Using(2),itcanbeshownthatT\nl5mf gandih\EJBC\r j fl5mf gIgJBC\r j lnmf 7 g 20204060801000.60.811.2nPSfragreplacements7e\nT\nT\n7e\nFig.1.Thisplotshowsthat andapproachand ,respectively,as\nincreases.Tosummarize,h\E'8h\E'\rT\n /T\n7e\n7 JBC\r j forlarge (3)Figure1showsaplotofT\n,7e\n,andT\n7e\nversus .For \r ,itseemsreasonabletouseT\nandtheapproximation(3)forthestandarderror.REFERENCES[1]E.L.LehmannandG.Casella,Theoryofpointestimation,Springer-Verlag,NewYork,1998.[2]M.Evans,N.Hastings,andB.Peacock,Statisticaldistributions,Wiley,NewYork,1993.[3]R.L.Graham,D.E.Knuth,andO.Patashnik,Concretemathematics:afoundationforcomputerscience,Addison-Wesley,Reading,1994.