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December, 1964 Convergence of the Losses of Certain Decision Rules for the Sequential Compound Decision Problem
Ester Samuel
Ann. Math. Statist. 35(4): 1606-1621 (December, 1964). DOI: 10.1214/aoms/1177700385


This paper is a continuation of [8], and considers the sequential compound decision problem for the case where the component decisions are of the simple versus simple hypothesis testing type, and thus can be stated in terms of testing whether $\theta = 0$ or $\theta = 1$. The loss for the compound decision is taken to be the average of the losses in the component decisions, and the risk for the compound decision is defined correspondingly. Let $R( )$ denote the Bayes envelope function of the component problem. In [8] two sequences of compound decision rules $\{T^\ast_n\}$ and $\{\hat T_n\}$ are exhibited, such that for $n$ sufficiently large, the risk incurred by $\hat{T}_n$ never exceeds $R(\vartheta_n) + \epsilon$ where $\vartheta_n$ is the average of the true $\theta$-values in the $n$ first components, and this holds uniformly in all possible sequences of $\theta$'s: for $T^\ast_n$ a corresponding statement is valid provided $R( )$ is differentiable for all $0 \leqq \eta \leqq 1$. Here we prove that for any sequence of $\theta$-values, the difference between the loss incurred by $\hat{T}_n$ and $R(\vartheta_n)$ converges to zero in probability, and under the differentiability assumption a corresponding statement holding with probability one is proved for $T^\ast_n$. Numerical data is provided to indicate the rate of convergence.


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Ester Samuel. "Convergence of the Losses of Certain Decision Rules for the Sequential Compound Decision Problem." Ann. Math. Statist. 35 (4) 1606 - 1621, December, 1964.


Published: December, 1964
First available in Project Euclid: 27 April 2007

zbMATH: 0134.35903
MathSciNet: MR181073
Digital Object Identifier: 10.1214/aoms/1177700385

Rights: Copyright © 1964 Institute of Mathematical Statistics

Vol.35 • No. 4 • December, 1964
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