Estimation and confidence intervals for empirical mixing distributions

Questions regarding collections of parameter estimates can frequently be expressed in terms of an empirical mixing distribution (EMD). This report discusses empirical Bayes estimation of an EMD, with emphasis on the construction of interval estimates. Estimation of the EMD is accomplished by substitution of estimates of prior parameters in the posterior mean of the EMD. This procedure is examined in a parametric model (the normal-normal mixture) and in a semi-parametric model. In both cases, the empirical Bayes bootstrap of Laird and Louis (1987, Journal of the American Statistical Association 82, 739-757) is used to assess the variability of the estimated EMD arising from the estimation of prior parameters. The proposed methods are applied to a meta-analysis of population trend estimates for groups of birds.