The Annals of Mathematical Statistics

An Essentially Complete Class of Admissible Decision Functions

Abraham Wald

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Abstract

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.

Article information

Source
Ann. Math. Statist. Volume 18, Number 4 (1947), 549-555.

Dates
First available in Project Euclid: 28 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aoms/1177730345

Digital Object Identifier
doi:10.1214/aoms/1177730345

Mathematical Reviews number (MathSciNet)
MR23499

Zentralblatt MATH identifier
0029.30604

JSTOR
links.jstor.org

Citation

Wald, Abraham. An Essentially Complete Class of Admissible Decision Functions. Ann. Math. Statist. 18 (1947), no. 4, 549--555. doi:10.1214/aoms/1177730345. http://projecteuclid.org/euclid.aoms/1177730345.


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