Some generalized subset selection procedures

Abstract

In this paper some generalizations of Gupta's subset selection procedure are discussed. Assume k(\geq 2) populations are given and assume that the associated random variables have distributions with unknown location parameters \theta_i, i = 1, ..., k. The ordered parameters are denoted by \theta_[1] \leq ... \leq \theta_[k] . On the basis of independent samples from these populations, Gupta (1965) selects a subset, as small as possible, which contains, with probability at least P*, the best population, i.e. the one with the largest location parameter, \theta_[k]. The two generalizations discussed in this paper are those of van der Laan (1991, 1992a, b) and of van der Laan and van Eeden (1993). Each one of these is designed to give a smaller expected subset size, ES, than Gupta's procedure, for which ES is large when \theta_[k] is close to the other \theta_i 's. The procedure of van der Laan (1992a) selects, with probability at least P*, an \epsilon-best population whose location parameter is at least \theta_[k] - \epsilon (with \epsilon \geq 0). Some efficiency results for normal populations, comparing van der Laan's procedure with Gupta's, are presented. The procedure of van der Laan and van Eeden (1993) uses a loss function and it upperbounds either the expected loss or the expected subset size, or both. The loss is taken as zero when the subset contains an \epsilon-best population and as an increasing function of \theta_[k] - \epsilon - max { \theta_i I i-th population in the subset } if not. Some properties of this procedure, for the case of two normal populations, are presented