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Bayes and likelihood calculations from confidence intervals. (English) Zbl 0773.62021
Summary: 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 $L\sp \dag\sb x(\theta)$ for $\theta$, with all nuisance parameters eliminated, based on any system of approximate confidence intervals. The statistician can then modify $L\sp \dag\sb x(\theta)$ 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 $L\sp \dag\sb x(\theta)$ based on this system requires only a few times as much computation as the maximum likelihood estimate $\widehat\theta$ and its estimated standard error $\widehat\sigma$. The formula for $L\sp \dag\sb x(\theta)$ is justified in terms of high-order adjusted likelihoods and also the Jeffreys-Welch and Peers [{\it B. L. Welch} and {\it H. W. Peers}, J. R. Stat. Soc., Ser. B 25, 318-329 (1963; Zbl 0117.142)] theory of uninformative priors. Several examples are given.

62F15Bayesian inference
62F10Point estimation
62F25Parametric tolerance and confidence regions
62C12Empirical decision procedures; empirical Bayes procedures
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