Smoothing the non-parametric estimate of a prior distribution by roughening. A computational study

Professor C.R. Rao has made foundational and applied contributions to simultaneous, or Empirical Bayes, inference. In these models data are generated by a compound process, where for each component under study parameters are sampled from a prior distribution, and then observations are drawn from a conditional distribution given the parameters. For example, in a multi-clinical medical trial, clinic-specific treatment effects θ₁,...,θ_K come from a distribution G, and then observed treatment effects are Gaussian with mean θ_k for the k th clinic. Inferential goals include point estimation and confidence intervals for individual parameters. Empirical Bayes methods have been developed to deal with the situation where G is not completely known. A flexible modeling approach bases inferences on a non-parametric estimate of G. Though the approach has attractive properties, this estimate is discrete and smoothing it may improve its empirical Bayes performance. In this research we investigate smoothing methods based on recursively "roughening" a smooth prior towards the discrete non-parametric estimate. Through mathematical analysis and numerical evaluations we show its performance in estimating G and producing empirical Bayes inferences.