Stability under contamination of robust regression estimators based on differences of residuals

Abstract

A reasonable approach to robust regression estimation is minimizing a robust scale estimator of the pairwise differences of residuals. We introduce a large class of estimators based on this strategy extending ideas of Yohai and Zamar (1993) and Croux, Rousseeuw and Hossjer (1994). The asymptotic robustness properties of the estimators in this class are addressed using the maxbias curve. We provide a general principle to compute this curve and present explicit formulae for several particular cases including generalized versions of S-, R- and '!-estimators. Finally, the most stable estimator in the class, that is, the estimator with the minimum maxbias curve, is shown to be the set of coefficients that minimizes an appropriate quantile of the distribution of the absolute pairwise differences of residuals.