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## 1 Introduction 

Consider a multivariate linear model (MLM) with random predictors, i.e., we observe n independent identically distributed (i.i.d.) ( p + q )-dimensional vec- tors, z$_{i}$ = ( y ′ $_{i}$, x ′ $_{i}$) with 1 ≤ i ≤ n , where y$_{i}$ = ( y$_{i}$$_{1}$, . . . , y$_{iq}$ ) ′ ∈ R $^{q}$, x$_{i}$ = ( x$_{i}$$_{1}$, . . . , x$_{ip}$ ) ′ ∈ R p and ′ denotes the transpose. The y$_{i}$ are the response vectors and the x$_{i}$ are the predictors and both satisfy the equation 

<img src='content_image/315068.jpg'>

where B$_{0}$ ∈ R p × q is the matrix of the regression parameters and u$_{i}$ is a q - dimensional vector independent of x$_{i}$ . If x$_{ip}$ = 1 for all 1 ≤ i ≤ n , we obtain a regression model with intercept. 

We denote the distributions of x$_{i}$ and u$_{i}$ by G$_{0}$ and F$_{0}$ , respectively, and Σ$_{0}$ is the covariance matrix of the u$_{i}$ . The p -multivariate normal distribution with mean vector µ and covariance matrix Σ is denoted by N$_{p}$ ( µ , Σ ). 

In the case of u$_{i}$ with distribution N$_{q}$ ( 0 , Σ$_{0}$ ), the maximum likelihood estimate (MLE) of B$_{0}$ is the least squares estimate (LSE), and the MLE of Σ$_{0}$ is the sample covariance matrix of the residuals. It is known that these estimates are not robust: a small fraction of outliers may have a large effect on their values. 

Several approaches have been proposed to deal with this problem. The first proposal of a robust estimate for the MLM was given Koenker and Portnoy [14]. They proposed to apply a regression M-estimator, based on a convex loss function, to each coordinate of the response vector. The problems with this estimate is lack of affine equivariance and zero breakdown point. Several other estimates without these problem were defined later. Rousseeuw et al. [22] proposed estimates for the MLM based on a robust estimate of the covariance matrix of z = ( x $^{′}$, y $^{′}$). 

