Estimating a Population of Parameter Values Using Bayes and Empirical Bayes Methods

Abstract

In standard Bayes and empirical Bayes component decision problems, estimating inidividual parameters is the primary goal. In multiple comparison problems and in comparisons of histograms of estimates, however, the primary goal is to produce parameter estimates that can be considered as an ensemble. For example, the histogram of estimates should be a good estimate of the histogram of parameters. Standard methods of estimating by the posterior expectation do minimize symmetric, componentwise losses such as squared error, but they produce ensembles of estimates with a sample variance smaller than the posterior expected sample variance for parameters. In this article we propose new Bayes and empirical Bayes estimates that minimize a distance function between the empirical cdf of the estimates and the true parameters. These estimators are weighted averages of the prior mean and the data, with weight on the data being approximately the square root of that for the posterior expectation. We give theoretical and applied examples, including subgroup analysis in a clinical trial.

Key Words:

• Bayesian methods
• Subgroup analysis
• Empirical Bayes
• Multiple comparisons