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Biometrika 1993 80(1):3-26; doi:10.1093/biomet/80.1.3
© 1993 by Biometrika Trust
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Bayes and likelihood calculations from confidence intervals

BRADLEY EFRON

Department of Statistics, Stanford University Stanford, California 94305-4065, U.S.A.

Recently there has been considerable progress on setting good approximate confidence intervals for a single parameter {theta} in a multi-parameter family. Here we use these frequentist results as a convenient device for making Bayes, empirical Bayes and likelihood inferences about {theta}. A simple formula is given that produces an approximate likelihood function Lx{dagger}({theta}) for {theta}, with all nuisance parameters eliminated, based on any system of approximate confidence intervals. The statistician can then modify Lx{dagger}({theta}) with Bayes or empirical Bayes information for {theta}, without worrying about nuisance parameters. The method is developed for multiparameter exponential families, where there exists a simple and accurate system of approximate confidence intervals for any smoothly defined parameter. The approximate likelihood Lx{dagger}({theta}) based on this system requires only a few times as much computation as the maximum likelihood estimate {theta} and its estimated standard error {sigma}. The formula for Lx{dagger}({theta}) is justified in terms of high-order adjusted likelihoods and also the Jeffreys-Welch & Peers theory of uninformative priors. Several examples are given.

Key Words: ABC interval • Bootstrap method • Empirical Bayes inference • Exponential family • Implied likelihood • Uninformative prior • Welch-Peers theory


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