Open Access
June 1998 Functional stability of one-step GM-estimators in approximately linear regression
Douglas G. Simpson, Victor J. Yohai
Ann. Statist. 26(3): 1147-1169 (June 1998). DOI: 10.1214/aos/1024691092

Abstract

This paper provides a comparative sensitivity analysis of one-step Newton–Raphson estimators for linear regression. Such estimators have been proposed as a way to combine the global stability of high breakdown estimators with the local stability of generalized maximum likelihood estimators. We analyze this strategy, obtaining upper bounds for the maximum bias induced by $\varepsilon$-contamination of the model. These bounds yield break-down points and local rates of convergence of the bias as $\varepsilon$decreases to zero. We treat a unified class of Newton–Raphson estimators, including one-step versions of the well-known Schweppe, Mallows and Hill–Ryan GM estimators. Of the three well-known types, the Hill–Ryan form emerges as the most stable in terms of one-step estimation. The Schweppe form is susceptible to a breakdown of the Hessian matrix. For this reason it fails to improve on the local stability of the initial estimator, and it may lead to falsely optimistic estimates of precision.

Citation

Download Citation

Douglas G. Simpson. Victor J. Yohai. "Functional stability of one-step GM-estimators in approximately linear regression." Ann. Statist. 26 (3) 1147 - 1169, June 1998. https://doi.org/10.1214/aos/1024691092

Information

Published: June 1998
First available in Project Euclid: 21 June 2002

zbMATH: 0930.62030
MathSciNet: MR1635458
Digital Object Identifier: 10.1214/aos/1024691092

Subjects:
Primary: 62F35 , 62J02
Secondary: 62F10 , 62J05

Keywords: Breakdown point , maximum bias function , Newton–Raphson , robust statistics , weighted least squares

Rights: Copyright © 1998 Institute of Mathematical Statistics

Vol.26 • No. 3 • June 1998
Back to Top