On selecting the pareto-optimal subset of a class of populations

Abstract

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 δ^*-Pareto-optimal if no vector in M, differing in at least one component from some mean vector μ corresponding to a population in G, has the property that each of its components is larger by at least δ^* > 0 than the corresponding component of the vector μ

In this paper we evaluate procedures devised to select the δ^*-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 or Larger values of k, we construct an algorithm and also some computer simulation to show how this can be done.