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On Selecting the Largest of k Normal Population Means

C. W. Dunnett
Journal of the Royal Statistical Society. Series B (Methodological)
Vol. 22, No. 1 (1960), pp. 1-40
Published by: Wiley for the Royal Statistical Society
Stable URL: http://www.jstor.org/stable/2983876
Page Count: 40
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On Selecting the Largest of k Normal Population Means
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Abstract

Consider a k-variate normal distribution with unknown means μ1,..., μk and with equal variances and equal covariances whose values are known; the problem is to select the variate associated with μmax 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 μ1,...,μk have a joint a priori normal distribution with means U1,...,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 μmax 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.

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