A note on the conditional probability of correct selection under the two-element partition

Abstract

Suppose there are k normal populations with unknown means, and the goal is to select the population having the largest mean based on independent observations from the k populations. Bechhofer (Ann. Math. Statist. 25 (1954), 16–39) proposed the indifference zone approach for this problem. The disadvantage of this approach is that often the same decision (which population to be selected) and the same probability of correct selection are assigned to two different sets of observations, one of which intuitively conveys a much stronger feeling of correct selection. Using Kiefer's (Proc. 4th Dayton Multivariate Conf., North-Holland, Amsterdam, 1975, 143–158; J. Amer. Statist. Assoc. 72 (1977), 789–827) conditional approach, the whole sample space can be partitioned into a set of lucky observations and a set of unlucky observations. In this note, we study the conditional probability of correct selection under this two-element partition. Results on least favourable configuration and ?-correct selection are established if we have lucky observations. Some unsolved problems are pointed out.