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December, 1947 An Essentially Complete Class of Admissible Decision Functions
Abraham Wald
Ann. Math. Statist. 18(4): 549-555 (December, 1947). DOI: 10.1214/aoms/1177730345


With any statistical decision procedure (function) there will be associated a risk function $r(\theta)$ where $r(\theta)$ denotes the risk due to possible wrong decisions when $\theta$ is the true parameter point. If an a priori probability distribution of $\theta$ is given, a decision procedure which minimizes the expected value of $r(\theta)$ is called the Bayes solution of the problem. The main result in this note may be stated as follows: Consider the class C of decision procedures consisting of all Bayes solutions corresponding to all possible a priori distributions of $\theta$. Under some weak conditions, for any decision procedure $T$ not in $C$ there exists a decision procedure $T^\ast$ in $C$ such that $r^\ast(\theta) \leqq r(\theta)$ identically in $\theta$. Here $r(\theta)$ is the risk function associated with $T$, and $r^\ast(\theta)$ is the risk function associated with $T^\ast$. Applications of this result to the problem of testing a hypothesis are made.


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Abraham Wald. "An Essentially Complete Class of Admissible Decision Functions." Ann. Math. Statist. 18 (4) 549 - 555, December, 1947.


Published: December, 1947
First available in Project Euclid: 28 April 2007

zbMATH: 0029.30604
MathSciNet: MR23499
Digital Object Identifier: 10.1214/aoms/1177730345

Rights: Copyright © 1947 Institute of Mathematical Statistics

Vol.18 • No. 4 • December, 1947
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