On the asymptotic efficiency of selection procedures for independent Gaussian populations (Q1689001)

The subject of the paper is a discrete event simulation and optimization technique related to the classic ranking and selection (R\&S) procedures when the number of populations is large. Applying insights from extreme value theory, the asymptotic behavior of linear combinations of maxima is specified. These results are used in order to derive new asymptotic results for R\&S procedures through the indifference-zone approach of \textit{R. E. Bechhofer} [Ann. Math. Stat. 25, 16--39 (1954; Zbl 0055.13003)]. Next, \textit{H. Robbins} and \textit{D. Siegmund}'s [in: Math. Decision Sci., Proc. 5th Summer Semin. Stanford 1967, Part 2, 267--279 (1968; Zbl 0211.50903)] problem of selecting from \(k\) independent normal homoscedastic populations with known variance by their means is generalized and, in addition, a new proof for the original result is presented. The procedures which were proposed respectively by \textit{E. J. Dudewicz} and \textit{S. R. Dalal} [Sankhyā, Ser. B 37, 28--78 (1975; Zbl 0337.62016)] and \textit{Y. Rinott} [Commun. Stat., Theory Methods A7, 799--811 (1978; Zbl 0392.62020)] are analyzed. Both procedures were designed for the problem of selecting the Gaussian population with the highest mean for independent populations with unknown and possibly different variances. First order approximations for these procedures asymptotic efficiencies are derived, measured in terms of the expected sample size required to achieve a desired probability for correct selection (PCS), as the total number of populations grows to infinity. Based on this approximation it is shown that asymptotically, Rinott's procedure is relatively less efficient than Dudewicz \& Dalal's one by a multiplicative factor depending on the initial sample size used in stage one of both procedures. A conjecture is formulated that the optimal sample size in the first stage of both procedures grows logarithmically in the number of populations, and with this optimal choice the multiplicative factor approaches one and the two procedures may be asymptotically equivalent.