An Essentially Complete Class of Admissible Decision Functions

With any statistical decision procedure (function) there will be associated a risk function $r( heta)$ where $r( heta)$ denotes the risk due to possible wrong decisions when $ heta$ is the true parameter point. If an a priori probability distribution of $ heta$ is given, a decision procedure which minimizes the expected value of $r( heta)$ 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 $ heta$. Under some weak conditions, for any decision procedure $T$ not in $C$ there exists a decision procedure $T^ast$ in $C$ such that $r^ast( heta) leqq r( heta)$ identically in $ heta$. Here $r( heta)$ is the risk function associated with $T$, and $r^ast( heta)$ is the risk function associated with $T^ast$. Applications of this result to the problem of testing a hypothesis are made.