On the global robustness of generalized S-estimators

Abstract

A generalized S-estimator (GS-estimator) of regression is obtained by minimizing an M-estimator of scale applied to the pairwise differences of residuals r i( θ)−r j( θ), i<j . In this paper, we focus on the global robustness properties of these estimators. It was pointed out by Croux et al. (J. Amer. Statist. Assoc. 89 (1994) 1271–1281) that two GS-estimators with the same breakdown point may have very different global robustness behavior. Accordingly, we supplement the information given by the breakdown point with the explosion rate, a summary measure of the robustness behavior of the estimator when the amount of contamination is large. We provide formulas that allow the computation of explosion rates and establish a link between the local behavior at zero of the score function on which the M-estimator is based and the robustness of the corresponding GS-estimator. Finally, we apply the explosion rate to quantify the loss of robustness when using GS-estimators instead of the simpler non-generalized S-estimators.