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

Thomas E. O'Bryan
The Annals of Statistics
Vol. 4, No. 6 (Nov., 1976), pp. 1290-1293
Stable URL: http://www.jstor.org/stable/2958598
Page Count: 4
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Some Empirical Bayes Results in the Case of Component Problems with Varying Sample Sizes for Discrete Exponential Families
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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 nth problem. Let θ = (θ1, θ2, ⋯) be a sequence of i.i.d. G random variables and let the conditional distribution of the observations Xn = (Xn,1, ⋯, Xn,m(n)) given θ be (Pθn )m(n), n = 1, 2, ⋯. For a decision concerning θn+1, where θ indexes a certain discrete exponential family, procedures tn are investigated which will utilize all the data X1, X2, ⋯, Xn+1 and which, under certain conditions, are asymptotically optimal in the sense that E|tn - θn+1|2 - Rm(n+1)(G) → 0 as n → ∞ for all G.

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