Approaches for empirical bayes confidence intervals

Parametric empirical Bayes (EB) methods of point estimation date to the landmark paper by James and Stein (1961). Interval estimation through parametric empirical Bayes techniques has a somewhat shorter history, which was summarized by Laird and Louis (1987). In the exchangeable case, one obtains a “naive” EB confidence interval by simply taking appropriate percentiles of the estimated posterior distribution of the parameter, where the estimation of the prior parameters (“hyperparameters”) is accomplished through the marginal distribution of the data. Unfortunately, these “naive” intervals tend to be too short, since they fail to account for the variability in the estimation of the hyperparameters. That is, they do not attain the desired coverage probability in the EB sense defined by Morris (1983a, b). They also provide no statement of conditional calibration (Rubin 1984). In this article we propose a conditional bias correction method for developing EM intervals that corrects these deficiencies in the naive intervals. As an alternative, several authors have suggested use of the marginal posterior in this regard. We attempt to clarify its role in achieving EB coverage. Results of extensive simulation of coverage probability and interval length for these approaches are presented in the context of several illustrative examples.