## The Annals of Statistics

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

### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176344849

**Digital Object Identifier**

doi:10.1214/aos/1176344849

**Mathematical Reviews number (MathSciNet)**

MR550153

**Zentralblatt MATH identifier**

0435.62026

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62F07: Ranking and selection

Secondary: 62G35: Robustness

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aos/1176344849