A Sample Reuse Method for Accurate Parametric Empirical Bayes Confidence Intervals

SUMMARY Parametric empirical Bayes (EB) methods of point estimation for a vector of unknown parameters date to the landmark paper of James and Stein (1961). The usual approach is to use the mean of the estimated posterior distribution of each parameter, where the estimation of the prior parameters (‘hyperparameters’) is accomplished through the marginal distribution of the data. While point estimates computed this way usually perform well, interval estimates based on the estimated posterior (called ‘naive’ EB intervals) do not. They fail to account for the variability in the estimation of the hyperparameters, generally resulting in subnominal coverage probability in the EB sense defined in Morris (1983a). In this paper we extend our earlier work in which we proposed a conditional bias correction method for developing EB intervals which corrects the deficiencies in the naive intervals. We show how bias correction can be implemented in general via a type III parametric bootstrap procedure, a sample reuse method first employed by Laird and Louis (1987). Theoretical and simulation work indicates that the resulting intervals are accurate with respect to nominal coverage. We give a specific application (to Poisson failure rate data) where we compute simultaneous point and bias-corrected interval estimates.