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Abstract:
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Confidence intervals for the means of multiple normal populations are often based on a hierarchical normal model. In this article we construct confidence intervals that have a constant frequentist coverage rate and that make use of information about across-group heterogeneity, resulting in constant-coverage intervals that are narrower than standard t-intervals on average across groups. Such intervals are constructed by inverting biased tests for the mean of a normal population. Given a prior distribution on the mean, Bayes-optimal biased tests can be inverted to form Bayes-optimal confidence intervals with frequentist coverage that is constant as a function of the mean. In the context of multiple groups, the prior distribution is replaced by a model of across-group heterogeneity. The parameters for this model can be estimated using data from all of the groups, and used to obtain confidence intervals with constant group-specific coverage that adapt to information about the distribution of group means.
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