ON SELECTING THE PARETO-OPTIMAL SUBSET OF A CLASS OF POPULATIONS

Let P be a class of k populations (with k known) each having an underlying multivariate normal distribution with unknown mean vector. We suppose that the mean (vector) value of each population can be represented by a vector parameter with p components and use the notation M to denote the set of all mean vectors for these k populations. Independent samples of size n are drawn from each population in P. We say that a subset G of P of vectors is delta*-Pareto-optimal if no vector in M, differing in at least one component from some mean vector-mu corresponding to a population in G, has the property that each of its components is larger by at least delta* > 0 than the corresponding component of the vector-mu. In this paper we evaluate procedures devised to select the delta*-Pareto-Optimal subset of a class of populations according to the minimum probability of correct selection over a region of the parameter space which we call the preference zone. For small values of k, theoretical calculations are given to analyze how big a sample size n is needed to bound the minimum probability of correct selection below by a preassigned value P*. We usually take P* close to 1 (say .90 or .95), although theoretically we only need to have P* > 1/(2k - 1). For larger values of k, we construct an algorithm and also some computer simulation to show how this can be done.