On the Relation between $S$-Estimators and $M$-Estimators of Multivariate Location and Covariance

Abstract

We discuss the relation between $S$-estimators and $M$-estimators of multivariate location and covariance. As in the case of the estimation of a multiple regression parameter, $S$-estimators are shown to satisfy first-order conditions of $M$-estimators. We show that the influence function IF $(\mathbf{x; S}, F)$ of $S$-functionals exists and is the same as that of corresponding $M$-functionals. Also, we show that $S$-estimators have a limiting normal distribution which is similar to the limiting normal distribution which is similar to the limiting normal distribution of $M$-estimators. Finally, we compare asymptotic variances and breakdown point of both types of estimators.