Asymptotics of Generalized S-Estimators

An S-estimator of regression is obtained by minimizing an M-estimator of scale applied to the residuals ri. On the other hand, a generalized S-estimator (or GS-estimator) minimizes an M-estimator of scale based on all pairwise differences ri − rj. Generalized S-estimators have similar robustness properties as S-estimators, including a high breakdown point. In this paper we prove asymptotic normality for the GS-esimator of the regression parameters, as well as for the accompanying scale estimator defined by the minimal value of the objective function. It turns out that the asymptotic efficiency can be much higher than that of S-estimators. For instance, by using a biweight ρ-function we obtain a GS-estimator with 50% breakdown point and 68.4% efficiency.