The Annals of Statistics

Most Economical Robust Selection Procedures for Location Parameters

S. R. Dalal and W. J. Hall

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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.

Article information

Ann. Statist., Volume 7, Number 6 (1979), 1321-1328.

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62F07: Ranking and selection
Secondary: 62G35: Robustness

Selection procedures robust procedures Huber's $M$-estimate location parameters uniform asymptotic normality


Dalal, S. R.; Hall, W. J. Most Economical Robust Selection Procedures for Location Parameters. Ann. Statist. 7 (1979), no. 6, 1321--1328. doi:10.1214/aos/1176344849.

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