Empirical Bayes Confidence Intervals Based on Bootstrap Samples.

Consider the model with data generated by the following two-stage process. First, a parameter theta is sampled from a prior distribution G, and then an observation is sampled from the conditional distribution f(y | theta). If the prior distribution is known, then the Bayes estimate under squared error loss is the posterior expectation of theta conditional on the data y. For example, if G is Gaussian with mean mu and variance tau[sup 2] and f(y | theta) is Gaussian with mean theta and variance sigma[sup 2], then the posterior distribution is Gaussian with mean beta mu + (1 - B)y and variance sigma[sup 2](1 - B), where B = sigma[sup 2]/(sigma[sup 2] + tau[sup 2]). Inferences about theta are based on this distribution. We study the application of the bootstrap to situations where the prior must be estimated from the data (empirical Bayes methods). For this model, we observe data Y[sup T] = [Y[sub 1], ..., Y[sub K]][sup T], each independent Y[sub k] following the compound model described previously. As first shown by James and Stein (1961), setting each. Theta[sub k] equal to its estimated posterior mean, (1 - B)Y[sub k] + B mu, where it = Y, B = min[1, (K - 3)sigma[sup 2]/S[sup 2], and S[sup 2] = SIGMA[sup K, sub k=1] (Y[sub k] - Y)[sup 2], produces estimates with smaller summed squared error loss than the maximum likelihood estimates theta[sub k] = Y[sub k]. In many applications, confidence intervals or other summaries are required, but computing them from the posterior based on an estimated prior (the naive approach) generally is inappropriate. These posterior distributions fail to account for the uncertainty in estimating the prior and, therefore, may be too compact or have an inappropriate shape. Several approaches have been proposed for incorporating this uncertainty, ranging from Bayes-empirical Bayes based on the introduction of a hyperprior (Deely and Lindley 1981), to use of the delta method (Morris 1983a,b). We develop and study bootstrap methods for... [ABSTRACT FROM AUTHOR]