Parametric Empirical Bayes Confidence Intervals

Abstract

This chapter outlines parametric empirical Bayes confidence intervals. Empirical Bayes modeling assumes the distributions π for the parameters θ= (θ1, …, θk) exist, with π taken from a known class Π of possible parameter distributions. Π is considered independent N (u, A) distributions on Rk. It is called parametric empirical Bayes problem, because πɛ Π is determined by the parameters (u, A) and so is a parametric family of distributions. A simulation presented in the chapter was used to determine that the intervals ▪ ±si and ▪ ±1.96si contain the true values θi in at least 68 percent and 95 percent of the cases. Empirical Bayes estimators, or Stein's estimator, can lead to misestimation of components that the statistician or his clients care about when exchangeability in the prior distribution is implausible. The term empirical Bayes, which is used for non-parametric empirical Bayes problems, actually fits the parametric empirical Bayes case too. The empirical Bayes methods in general and parametric empirical Bayes methods in particular provide a way to utilize this additional information by obtaining more precise estimates and estimating their precision.