On the relation between S-estimators and M-estimators of multivariate location and covariance

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 (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.