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.