A Classification Problem in which Information about Alternative Distributions is Based on Samples

Abstract

The problem of classification in the case of a finite number of known distributions has been considered many times in the statistical literature, and the theory is well-developed for such problems. Some authors have considered the problem of classification in the case where some of the information about the alternative distributions has been obtained from samples. Papers concerned with this latter problem usually either present large-sample results, or else propose procedures whose use in the small-sample case is justified on intuitive or heuristic grounds (see, e.g., [4], [5], [6]). In this paper, a certain classification problem is considered in which some of the information about the alternative multivariate normal distributions has been obtained from samples. The admissibility of two "natural" decision procedures is deduced. Charles Stein has shown in [7] that "natural" procedures are not necessarily admissible when one is dealing with multivariate normal distributions. The problem is defined in Section two. In Section three, several heuristic methods of solution are considered. Each of these methods yields one or the other of two decision procedures which are called the minimum distance rule, and the restricted maximum likelihood rule, respectively. In Section four, a method for obtaining admissible translation-invariant Bayes procedures is presented. This method consists in a reparametrization, and the use of an a priori distribution of the new parameters which has a certain product measure form. In Section five, normal a priori distributions are employed to obtain a particular class of translation-invariant Bayes procedures. The results of Section five are used in Section six to show that both the minimum distance rule and the restricted maximum likelihood rule are admissible translation=invariant Bayes procedures.