Most Economical Robust Selection Procedures for Location Parameters

Abstract

Consider samples of size $n$ from each of $k$ symmetric populations, differing only in their location parameters. The decision problem is to select the best population--the one with the largest location parameter--with control on the probability of correct selection (PCS) whenever the largest parameter is at least $\Delta$ units larger than all others, and whenever the common error distribution belongs to a specified neighborhood of the standard normal. It is shown that, if the sample size $n$ is chosen according to a formula given herein, and Huber's $M$-estimate is applied to each of the $k$ samples with the population having the largest estimate being selected as best, that the PCS goal is achieved asymptotically (as $\Delta\downarrow 0$)--the procedure is robust. Moreover, no other selection procedure can achieve this goal asymptotically with a smaller sample size--the procedure is most economical. Comparisons with other procedures are given. These results are based on a uniform asymptotic normality theorem for Huber's $M$-estimate, contained herein.