Mathematical Statistics - Jun Shaochap6

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1、Chapter6HypothesisTestsAgeneraltheoryoftestinghypothesesispresentedinthischapter.LetXbeasamplefromapopulationPinP,afamilyofpopulations.BasedontheobservedX,wetestagivenhypothesisH0:P∈P0versusH1:P∈P1,whereP0andP1aretwodisjointsubsetsofPandP0∪P1=P.Notationalconventionsandbasicc

2、oncepts(suchastwotypesoferrors,significancelevels,andsizes)giveninExample2.20and§2.4.2areusedinthischapter.6.1UMPTestsAtestforahypothesisisastatisticT(X)takingvaluesin[0,1].WhenX=xisobserved,werejectH0withprobabilityT(x)andacceptH0withprobability1−T(x).IfT(X)=1or0a.s.P,thenT(

3、X)isanonrandomizedtest.OtherwiseT(X)isarandomizedtest.ForagiventestT(X),thepowerfunctionofT(X)isdefinedtobeβT(P)=E[T(X)],P∈P,(6.1)whichisthetypeIerrorprobabilityofT(X)whenP∈P0andoneminusthetypeIIerrorprobabilityofT(X)whenP∈P1.Aswediscussedin§2.4.2,withasampleofafixedsize,weare

4、notabletominimizetwoerrorprobabilitiessimultaneously.OurapproachinvolvesmaximizingthepowerβT(P)overallP∈P1(i.e.,minimizingthetypeIIerrorprobability)andoveralltestsTsatisfyingsupβT(P)≤α,(6.2)P∈P0whereα∈[0,1]isagivenlevelofsignificance.Recallthattheleft-handsideof(6.2)isdefinedt

5、obethesizeofT.3933946.HypothesisTestsDefinition6.1.AtestT∗ofsizeαisauniformlymostpowerful(UMP)testifandonlyifβT∗(P)≥βT(P)forallP∈P1andToflevelα.IfU(X)isasufficientstatisticforP∈P,thenforanytestT(X),E(T

6、U)hasthesamepowerfunctionasTand,therefore,tofindaUMPtestwemayconsiderteststha

7、tarefunctionsofUonly.TheexistenceandcharacteristicsofUMPtestsarestudiedinthissec-tion.6.1.1TheNeyman-PearsonlemmaAhypothesisH0(orH1)issaidtobesimpleifandonlyifP0(orP1)containsexactlyonepopulation.Thefollowingusefulresult,whichhasalreadybeenusedonceintheproofofTheorem4.16,pro

8、videstheformofUMPtestswhenbothH0andH1aresimple.Theorem6.1(Neyman-Pearsonlemma).SupposethatP0={P0}andP1={P1}.Letfjbethep.d.f.ofPjw.r.t.aσ-finitemeasureν(e.g.,ν=P0+P1),j=0,1.(i)(ExistenceofaUMPtest).Foreveryα,thereexistsaUMPtestofsizeα,whichisequalto1f1(X)>cf0(X)T∗(X)=γf1(X)=

9、cf0(X)(6.3)0f1(X)