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1、Chapter7Con?denceSetsVariousmethodsofconstructingcon?dencesetsareintroducedinthischap-ter,alongwithstudiesofpropertiesofcon?dencesets.ThroughoutthischapterX=(X1,...,Xn)denotesasamplefromapopulationP∈P;θ=θ(P)denotesafunctionalfromPtoΘ?Rkfora?xedintegerk;andC(X)denotesa
2、con?dencesetforθ,asetinBΘ(theclassofBorelsetsonΘ)dependingonlyonX.Weadoptthebasicconceptsofcon?dencesetsintroducedin§2.4.3.Inparticular,infP∈PP(θ∈C(X))isthecon?dencecoe?cientofC(X)and,ifthecon?dencecoe?cientofC(X)is≥1?αfor?xedα∈(0,1),thenwesaythatC(X)hassigni?cancelev
3、el1?αorC(X)isalevel1?αcon?denceset.7.1ConstructionofCon?denceSetsInthissection,weintroducesomebasicmethodsforconstructingcon?dencesetsthathaveagivensigni?cancelevel(orcon?dencecoe?cient)forany?xedn.Propertiesandcomparisonsofcon?dencesetsaregivenin§7.2.7.1.1Pivotalquan
4、titiesPerhapsthemostpopularmethodofconstructingcon?dencesetsistheuseofpivotalquantitiesde?nedasfollows.De?nition7.1.AknownBorelfunction?of(X,θ)iscalledapivotalquantityifandonlyifthedistributionof?(X,θ)doesnotdependonP.NotethatapivotalquantitydependsonPthroughθ=θ(P).Ap
5、ivotalquantityisusuallynotastatistic,althoughitsdistributionisknown.4714727.Con?denceSetsWithapivotalquantity?(X,θ),alevel1?αcon?dencesetforanygivenα∈(0,1)canbeobtainedasfollows.First,?ndtwoconstantsc1andc2suchthatP(c1≤?(X,θ)≤c2)≥1?α.(7.1)Next,de?neC(X)={θ∈Θ:c1≤?(X,θ)
6、≤c2}.(7.2)ThenC(X)isalevel1?αcon?denceset,since
