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J.O. Berger 25923.5 Conditional frequentist inference23.5.1 IntroductionThe theory of combining the frequentist principle with conditioning was formalizedby Kiefer in Kiefer (1977), although there were many precursors to thetheory initiated by Fisher and others. There are several versions of the theory,but the most useful has been to begin by defining a conditioning statistic Swhich measures the “strength of evidence” in the data. Then one computesthe desired frequentist measure, but does so conditional on the strength ofevidence S.Pedagogical example continued: S = |X 1 − X 2 | is the obvious choice,S =2reflectingdatawithmaximalevidentialcontent(correspondingtothesituation of 100% confidence) and S = 0 being data of minimal evidentialcontent. Here coverage probability is the desired frequentist criterion, and aneasy computation shows that conditional coverage given S is given byP θ {C(X 1 ,X 2 )containsθ | S =2} = 1,P θ {C(X 1 ,X 2 )containsθ | S =0} = 1/2,for the two distinct cases, which are the intuitively correct answers.23.5.2 Ancillary statistics and invariant modelsAn ancillary statistic is a statistic S whose distribution does not depend onunknown model parameters θ. Inthepedagogicalexample,S =0andS =2have probability 1/2 each, independent of θ, and so S is ancillary. When ancillarystatistics exist, they are usually good measures of the strength of evidencein the data, and hence provide good candidates for conditional frequentist inference.The most important situations involving ancillary statistics arise when themodel has what is called a group-invariance structure; cf. Berger (1985) andEaton (1989). When this structure is present, the best ancillary statistic to useis what is called the maximal invariant statistic. Doing conditional frequentistinference with the maximal invariant statistic is then equivalent to performingBayesian inference with the right-Haar prior distribution with respect to thegroup action; cf. Berger (1985), Eaton (1989), and Stein (1965).Example–Location Distributions: Suppose X 1 ,...,X n form a randomsample from the location density f(x i − θ). This model is invariant under thegroup operation defined by adding any constant to each observation and θ; themaximal invariant statistic (in general) is S =(x 2 − x 1 ,x 3 − x 1 ,...,x n − x 1 ),and performing conditional frequentist inference, conditional on S, willgivethe same numerical answers as performing Bayesian inference with the right-