An Essentially Complete Class of Admissible Decision Functions

Abstract

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 {B}ayes 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 {B}ayes 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.