Selection of the Best Normal Population: A New Exact Solution, Asymptotically Optimal When k=2

Abstract

SYNOPTIC ABSTRACTThe problem we deal with in this paper was first stated in 1954 in Bechhofer's pioneering paper in ranking and selection. Namely, there are k (2 or more) normal populations with all parameters unknown, and the goal is to select one population which has the largest mean. Despite much interest (and a number of failed attempts to solve this problem in the literature), the first solution came in a 1975 paper by Dudewicz and Dalal. A slightly different solution followed in a 1978 paper by Rinott. While there has been other work on the problem, to date these are the only exact solutions (all other “solutions” are either heuristic or asymptotic only). It has been recognized for some time that the available exact solutions do not (asymptotically) allocate sample size in a way which corresponds to what is known to be optimal when the variances are known. In this paper we give a new, third, exact solution to the problem. The new solution is asymptotically optimal in the case when there are two populations. In an example on a real data set, we illustrate the procedure and see that it would save 7% of the sample size in that example (and asymptotically saves 6% in that example).