Optimal Confidence Interval for the Largest Normal Mean

Abstract

A two-stage sampling procedure is used to obtain an optimal confidence interval for the largest or smallest mean of more than two independent normal populations, where the population variances are unknown and possibly unequal. The optimal confidence interval is obtained by maximizing the coverage probability with a fixed width at a least favorable configuration of means over an overall parameter space. It will be shown that the optimal interval is uniformly better than any existing intervals in its class. Furthermore, the connection to a simpler one-stage sampling procedure is elaborated and a numerical example is given.