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Robust M Estimation in the Two-Sample Problem

Michael G. Akritas
Journal of the American Statistical Association
Vol. 86, No. 413 (Mar., 1991), pp. 201-204
DOI: 10.2307/2289731
Stable URL: http://www.jstor.org/stable/2289731
Page Count: 4
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Robust M Estimation in the Two-Sample Problem
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Abstract

Let X11, ..., X1n1 and X21, ..., X2n2 be two independent samples from a location-scale family, and suppose that the interest lies in robust M estimation of the location difference θ2 - θ1. It will be assumed that the two scale parameters are equal but unknown. The usual robust M estimator for the location shift is the difference of the robust M estimators for the two location parameters. In this article we propose an alternative method that uses a single estimating equation to generate the robust estimator of the location shift. This estimating equation is derived from a reparametrization of the two-sample problem introduced by Neyman and Scott. When the same ψ function is used, the new and the usual M estimators of the location shift have the same influence functions and asymptotic variance, and hence the two types of M estimators share the same class of optimal ψ functions. However, all members of the new class of optimal robust M estimators for the location shift have sample-wise breakdown point .5 and thus are preferable over the usual M estimators in the asymmetric case. (In the symmetric case the usual M estimators also have sample-wise breakdown point .5.) The essential idea is to use the orthogonality of the Neyman-Scott parametrization, and then to use a maximum breakdown point (but possibly low-efficiency) estimator for the nuisance parameter. Some theoretical breakdown points and numerical illustrations are given.

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