## The Annals of Mathematical Statistics

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

### An Essentially Complete Class of Admissible Decision Functions

#### 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.