  We consider the robust estimation of the parameters of multivariate Gaussian
linear regression models. To this aim we consider robust version of the usual
(Mahalanobis) least-square criterion, with or without Ridge regularization. We
introduce two methods each considered contrast: (i) online stochastic gradient
descent algorithms and their averaged versions and (ii) offline fix-point
algorithms. Under weak assumptions, we prove the asymptotic normality of the
resulting estimates. Because the variance matrix of the noise is usually
unknown, we propose to plug a robust estimate of it in the Mahalanobis-based
stochastic gradient descent algorithms. We show, on synthetic data, the
dramatic gain in terms of robustness of the proposed estimates as compared to
the classical least-square ones. Well also show the computational efficiency of
the online versions of the proposed algorithms. All the proposed algorithms are
implemented in the R package RobRegression available on CRAN.
