A Restricted Subset Selection Approach to Ranking and Selection Problems

Abstract

Let $\pi_1,\cdots, \pi_k$ be $k$ populations with $\pi_i$ characterized by a scalar $\lambda_i \in \Lambda$, a specified interval on the real line. The object of the problem is to make some inference about $\pi_{(k)}$, the population with largest $\lambda_i$. The present work studies rules which select a random number of populations between one and $m$ where the upper bound, $m$, is imposed by inherent setup restrictions of the subset selection and indifference zone approaches. A selection procedure is defined in terms of a set of consistent sequences of estimators for the $\lambda_i$'s. It is proved the infimum of the probability of a correct selection occurs at a point in the preference zone for which the parameters are as close together as possible. Conditions are given which allow evaluation of this last infimum. The number of non-best populations selected, the total number of populations selected, and their expectations are studied both asymptotically and for fixed $n$. Other desirable properties of the rule are also studied.