On Selecting the Largest of k Normal Population Means

Summary Consider a k-variate normal distribution with unknown means μ 1, …, μk and with equal variances and equal covariances whose values are known; the problem is to select the variate associated with μ max using a single-sample procedure of size n, and, in particular, to determine the required value of n. Previous work is reviewed. New results are given assuming that the population means μ 1, …, μk have a joint a priori normal distribution with means U 1, …, Uk known up to an additive constant, and with equal variances and equal covariances whose values are known. Results are given first for the probability of correct selection, then for that probability taken conditionally on the largest μ exceeding the other μ’s by at least a specified amount, and for the probability of the selected mean being within a specified amount of the largest, the latter formulation being equivalent to considering a certain zero-one loss function. Next a loss function is assumed linear in the difference between μ max and the selected μ and an experimentation cost proportional to n. The choice of k and n is considered and the relation with the plant-selection problem pointed out. The minimax solution and the concept of the “largest admissible” value of n are also discussed. In an appendix an expression is given for the expected value of the maximum of k equally correlated normal variates with equal variances but unequal means.