An Essentially Complete Class of Admissible Decision Functions

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