Mathematical Statistics - Jun Shaochap7

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1、Chapter7ConfidenceSetsVariousmethodsofconstructingconfidencesetsareintroducedinthischap-ter,alongwithstudiesofpropertiesofconfidencesets.ThroughoutthischapterX=(X1,...,Xn)denotesasamplefromapopulationP∈P;θ=θ(P)denotesafunctionalfromPtoΘ⊂Rkforafixedintegerk;andC(X)denotesa

2、confidencesetforθ,asetinBΘ(theclassofBorelsetsonΘ)dependingonlyonX.Weadoptthebasicconceptsofconfidencesetsintroducedin§2.4.3.Inparticular,infP∈PP(θ∈C(X))istheconfidencecoefficientofC(X)and,iftheconfidencecoefficientofC(X)is≥1−αforfixedα∈(0,1),thenwesaythatC(X)hassignificancelev

3、el1−αorC(X)isalevel1−αconfidenceset.7.1ConstructionofConfidenceSetsInthissection,weintroducesomebasicmethodsforconstructingconfidencesetsthathaveagivensignificancelevel(orconfidencecoefficient)foranyfixedn.Propertiesandcomparisonsofconfidencesetsaregivenin§7.2.7.1.1Pivotalquan

4、titiesPerhapsthemostpopularmethodofconstructingconfidencesetsistheuseofpivotalquantitiesdefinedasfollows.Definition7.1.AknownBorelfunctionℜof(X,θ)iscalledapivotalquantityifandonlyifthedistributionofℜ(X,θ)doesnotdependonP.NotethatapivotalquantitydependsonPthroughθ=θ(P).Ap

5、ivotalquantityisusuallynotastatistic,althoughitsdistributionisknown.4714727.ConfidenceSetsWithapivotalquantityℜ(X,θ),alevel1−αconfidencesetforanygivenα∈(0,1)canbeobtainedasfollows.First,findtwoconstantsc1andc2suchthatP(c1≤ℜ(X,θ)≤c2)≥1−α.(7.1)Next,defineC(X)={θ∈Θ:c1≤ℜ(X,θ)

6、≤c2}.(7.2)ThenC(X)isalevel1−αconfidenceset,since

infPθ∈C(X)=infPc1≤ℜ(X,θ)≤c2P∈PP∈P
=Pc1≤ℜ(X,θ)≤c2≥1−α.NotethattheconfidencecoefficientofC(X)maynotbe1−α.Ifℜ(X,θ)hasacontinuousc.d.f.,thenwecanchooseci’ssuchthattheequalityin(7.1)holdsand,therefore,theconfidencesetC(X)hasc

7、onfidencecoefficient1−α.Inagivenproblem,theremaynotexistanypivotalquantity,ortheremaybemanydifferentpivotalquantities.Whentherearemanypivotalquantities,onehastochooseonebasedonsomeprinciplesorcriteria,whicharediscussedin§7.2.Forexample,pivotalquantitiesbasedonsufficientstat

8、isticsarecertainlypreferred.Inmanycaseswealsohavetochooseci’sin(7.1)basedonsomecriteria.Whenℜ(X,θ)andci’sarechosen,weneedtoc