Most Economical Multiple-Decision Rules

Abstract

This paper is concerned with non-sequential multiple-decision procedures for which the sample size is a minimum subject to either (1) lower bounds on the probabilities of making correct decisions or (2) upper bounds on the probabilities of making incorrect decisions. Such decision procedures are obtained by constructing artificial decision problems for which the minimax strategies provide solutions to problems (1) and (2). These are shown to be "likelihood ratio" and "unlikelihood ratio" decision rules, respectively. Thus, although problems (1) and (2) are formulated in the spirit of the classical Neyman-Pearson approach to two-decision problems, minimax theory is used as a tool for their solution. Problems of both "simple" and "composite" discrimination are considered and some examples indicated. (Some multivariate examples are given in [4].) Various properties of the decision rules are derived, and relationships with works of Wald, Lindley, Rao and others are cited.