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.
@article{1176347386,
author = {Lopuhaa, Hendrik P.},
title = {On the Relation between $S$-Estimators and $M$-Estimators of Multivariate Location and Covariance},
journal = {Ann. Statist.},
volume = {17},
number = {1},
year = {1989},
pages = { 1662-1683},
language = {en},
url = {http://dml.mathdoc.fr/item/1176347386}
}
Lopuhaa, Hendrik P. On the Relation between $S$-Estimators and $M$-Estimators of Multivariate Location and Covariance. Ann. Statist., Tome 17 (1989) no. 1, pp. 1662-1683. http://gdmltest.u-ga.fr/item/1176347386/