Robust M estimation in the two-sample problem

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

Let (Equation presented) 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.

Original languageEnglish (US)
Pages (from-to)201-204
Number of pages4
JournalJournal of the American Statistical Association
Volume86
Issue number413
DOIs
StatePublished - Jan 1 1991

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M-estimation
Two-sample Problem
M-estimator
Breakdown Point
Estimating Equation
Two Parameters
Location-scale Family
Reparametrization
Influence Function
Robust Estimators
Location Parameter
Nuisance Parameter
Asymptotic Variance
Scale Parameter
Orthogonality
Parametrization
Estimator
Unknown
Breakdown
Alternatives

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

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abstract = "Let (Equation presented) 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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Robust M estimation in the two-sample problem. / Akritas, Michael G.

In: Journal of the American Statistical Association, Vol. 86, No. 413, 01.01.1991, p. 201-204.

Research output: Contribution to journalArticle

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AB - Let (Equation presented) 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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