Some Empirical Bayes Results in the Case of Component Problems with Varying Sample Sizes for Discrete Exponential Families

Abstract

Consider a modified version of the empirical Bayes decision problem where the component problems in the sequence are not identical in that the sample size may vary. In this case there is not a single Bayes envelope $R(\bullet)$, but rather a sequence of envelopes $R^{m(n)}(\bullet)$ where $m(n)$ is the sample size in the $n$th problem. Let $\mathbf{\theta} = (\theta_1, \theta_2, \cdots)$ be a sequence of i.i.d. $G$ random variables and let the conditional distribution of the observations $\mathbf{X}_n = (X_{n,1}, \cdots, X_{n,m(n)})$ given $\mathbf{\theta}$ be $(P_{\theta_n})^{m(n)}, n = 1, 2, \cdots$. For a decision concerning $\theta_{n+1}$, where $\theta$ indexes a certain discrete exponential family, procedures $t_n$ are investigated which will utilize all the data $\mathbf{X}_1, \mathbf{X}_2, \cdots, \mathbf{X}_{n+1}$ and which, under certain conditions, are asymptotically optimal in the sense that $E|t_n - \theta_{n+1}|^2 - R^{m(n+1)}(G) \rightarrow 0$ as $n \rightarrow \infty$ for all $G$.