parametric sequential confidence estimation

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1、SequentialEstimationbyMalayGhosh,NitisMukhopadhyayandPranabK.SenCopyright©1997JohnWiley&Sons,Inc.Chapter8ParametricSequentialConfidenceEstimation8.1INTRODUCTIONThechapterbeginswiththeproblemoffindingaconfidenceintervalforthenormalmeanwithfixedwidthandprescribedconfidence.AsdiscussedinChapters3and6,t

2、hisproblemcannotbesolvedwithafixedsamplesize.Stein'stwo-stageprocedureachievestheobjective,buttheASNforStein'scanbemuchlargerthantheoptimalsamplesizemeetingthesamegoalwithknownpopulationvariance.TheusualcriticismleveledagainstStein'sprocedureisthatitdoesnotuseanupdatedestimateofthevarianceindefining

3、thestoppingrule.Thefirstsequentialprocedureusinganupdatedestimateofthevariancewaspro-posedbyRay(1957).Ray'sprocedureisdescribedinSection8.2,andtheinaccuracyofhisconclusionispointedout.AlsothissectionintroducestheChow-Robbinsprocedureandstudiesitsproperties.InSection8.3wedis-cusssequentialintervalest

4、imationofthedifferenceoftwonormalmeanswithpossiblyunknownandunequalvariances.Section8.4dealswithfixedsizeconfidenceboundsforlinearregressionparameters.Finally,inSection8.5,wediscussconfidenceregionsforthemultivariatenormalmean.8.2FIXED-WIDTHINTERVALESTIMATIONOFTHENORMALMEANWebeginwiththeprocedureofR

5、ay(1957).ConsiderasequenceΧχ,Χι,...ofi.i.d.Ν(μ,σ2)variables,whereμ(real)isunknown.Foragivend(>0),recall(fromChapter6)thatifoneproposestheconfidenceinterval[Xn—d,Xn+d]forμ(Xn=n-1£"_,^"n>')>tnen*°Γknowner2,theoptimalsamplesizerequiredforattainingthepreassignedcoverageprobability1—a(0

6、positiveinteger>a2a2/d2=n*(say);aistheupper100(a/2)%pointofiV(0,1).Forsimplicity,weassumen*tobeapositiveinteger.2492508.PARAMETRICSEQUENTIALCONFIDENCEESTIMATIONTable8.1.n0=3,a=1.96A.51.01.2522.5c'(A).99975.9775.941.931.929¿>*(A)4.16.68.416.724.9ThestoppingruleproposedbyRayisgivenbelow.Welets2=(n—l)-

7、1Σ,7=ι(χ*-Xn)*(n>2)andN*=thesmallestoddintegern>n0(>3)forwhichn>a2»*«/-'.Usetheconfidenceinterval[ΧΝ'—d,Xff+d]forμ.Inordertostudythecoverageprobabilityoftheproposedstoppingrule,defineA=σ2/«*2,C*(A)=Ρμ