Minimax Solution of Statistical Decision Problems by Iteration

Abstract

For the general statistical decision problem, Wald suggested the minimax solution as one possible choice of an optimum solution. Sufficient conditions for the existence of minimax solutions are well known. However, minimax solutions have been calculated only for isolated problems. A drawback of the methods that have been used to obtain minimax solutions is that there is no guarantee that they will yield a minimax solution when they are applied to a specific new problem. An exception to this is M. N. Ghosh's method [2] of approximating minimax estimators for a certain class of problems. Since his method differs from the method presented here, no further mention of his work will be made. Presented in Section 3 is an iterative method of calculating minimax solutions that is applicable to a class of problems. The method is based on the result that, under certain conditions, if, for a sequence of a priori distributions on the parameter space, the corresponding sequence of Bayes risks converges to the supremum of all Bayes risks, then the corresponding sequence of Bayes decision functions converges to the minimax decision function and the corresponding sequence of risk functions converges uniformly to the minimax risk function. A method for iteratively constructing such a sequence of a priori distributions is given. The derivation in Section 3 of the iterative method of calculating minimax solutions depends on some well-known results of Wald [5]. These results are stated in Section 2 for convenience of reference and are proved for the sake of completeness, although many of the proofs are available in [5] or elsewhere. Also, previously no exposition of the results of Section 2 was available in the literature with the mathematical presentation used here. The assumptions used here to obtain Wald's results are weaker than those used in [5] and are similar to Wald's Assumptions 5.1 through 5.6 in [6]. However, in [6] Wald did not give all of the results of Section 2. Also, the results hold for a larger class of problems than Wald indicated. Wald's restriction to fixed sample size problems is unnecessary; the results apply to sequential problems with prescribed sequential experimentation rule of the type considered by Wald in [6]. In [6], Wald treats discrete and continuous random variables separately. Here the results are given in a generality that includes discrete and continuous random variables as special cases.