A Restricted Subset Selection Rule for Selecting At Least One of the t Best Normal Populations in Terms of Their Means: Common Known Variance Case

Abstract

Consider k (≥ 2) normal populations with unknown means μ₁,..., μ_k and a common known variance σ². Let μ_[1] ≤ ... ≤ μ_[k] denote the ordered μ_i. Our goal is to select a non-empty subset of the k populations whose size is at most m (1 ≤ m ≤ k − t) so that at least one of the populations associated with the t (1 ≤ t ≤ k − 1) largest means is included in the selected subset with a minimum guaranteed probability P*, whenever μ_[k−t+1] − μ_[k−t] ≥ δ* where P* and δ* are specified in advance of the experiment. Santner (1976) proposed and investigated a procedure (R _s) based on samples of size n from each of the populations. We propose and investigate an alternative procedure R _hp with the same sampling scheme. We compare our rule with that of a procedure that selects a subset of fixed size m. The special case of t = 1 was earlier studied by Gupta and Santner (1973) and Hsu and Panchapakesan (2003).