A restricted subset selection procedure for selecting the largest normal mean under heteroscedasticity

This article considers the goal of selecting the population with the largest mean among k normal populations when variances are not known. We propose a Stein-type two-sample procedure, denoted by (Formula presented.) for selecting a nonempty random-size subset of size at most m ((Formula presented.)) that contains the population associated with the largest mean, with a guaranteed minimum probability (Formula presented.) whenever the distance between the largest mean and the second largest mean is at least (Formula presented.) where m, (Formula presented.) and (Formula presented.) are specified in advance of the experiment. The probability of a correct selection and the expected subset size of (Formula presented.) are derived. Critical values/procedure parameters that are required for certain k, m, (Formula presented.) and (Formula presented.) are obtained by solving simultaneous integral equations and are presented in tables.