On Minimax Statistical Decision Procedures and their Admissibility

Abstract

This paper is concerned with the problem of making a decision on the basis of a sequence of observations on a random variable. Two loss functions, each depending on the distribution of the random variable, the number of observations taken, and the decision made, are assumed given. Minimax problems can be stated for weighted sums of the two loss functions, or for either one subject to an upper bound on the expectation of the other. Under suitable conditions it is shown that solutions of the first type of problem provide solutions for all problems of the latter types, and that admissibility for a problem of the first type implies admissibility for problems of the latter types. Two examples are given: Estimation of the mean of a random variable which is (1) normal with known variance, (2) rectangular with known range. The resulting minimax estimates are, with a small class of exceptions, proved admissible among the class of all procedures with continuous risk functions. The two loss functions are in each case the number of observations, and an arbitrary nondecreasing function of the absolute error of estimate. Extensions to a function of the number of observations for the first loss function are indicated, and two examples are given for the normal case where the sample size can or must be randomised among more than a consecutive pair of integers.