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November, 1976 Some Empirical Bayes Results in the Case of Component Problems with Varying Sample Sizes for Discrete Exponential Families
Thomas E. O'Bryan
Ann. Statist. 4(6): 1290-1293 (November, 1976). DOI: 10.1214/aos/1176343661

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 $G$.

Citation

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Thomas E. O'Bryan. "Some Empirical Bayes Results in the Case of Component Problems with Varying Sample Sizes for Discrete Exponential Families." Ann. Statist. 4 (6) 1290 - 1293, November, 1976. https://doi.org/10.1214/aos/1176343661

Information

Published: November, 1976
First available in Project Euclid: 12 April 2007

zbMATH: 0346.62007
MathSciNet: MR443155
Digital Object Identifier: 10.1214/aos/1176343661

Subjects:
Primary: 62C99
Secondary: 62C25 , 62C99

Keywords: decision theory , Empirical Bayes , squared error loss estimation

Rights: Copyright © 1976 Institute of Mathematical Statistics

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Vol.4 • No. 6 • November, 1976
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