A two-stage procedure for selecting the largest normal mean whose first stage selects a bounded random number of populations

Abstract

Abstract Suppose ∏ 1 ,…, ∏ k are normal populations with unknown means μ 1 ,…, μ k , respectively, and common known variance σ 2 . We study a class of two-stage procedures for selecting the normal population with the largest mean. These procedures screen populations indicated as being inferior in the first stage and select a single population at their terminal decision. The screening stage chooses a random number of populations for further study and also has the flexibility that the experimenter can specify the maximum number of populations entering the second stage. The second stage selects a single population from those not eliminated in the first-stage as the best one. A lower bound is determined for the probability of correct selection (PCS) which approximates the PCS closely. Subject to achieving the indifference zone probability requirement of Bechhofer (1954), an optimal design problem is formulated for determining the sample sizes and constants used to define the procedure. An approximate conservative solution is implemented for the design problem based on the lower bound for the PCS. Comparisons of the procedure so derived are made with the single-stage procedure of Bechhofer (1954) and the two-stage procedure of Tamhane and Bechhofer (1977,1979).