  Robust estimators of large covariance matrices are considered, comprising
regularized (linear shrinkage) modifications of Maronna's classical
M-estimators. These estimators provide robustness to outliers, while
simultaneously being well-defined when the number of samples does not exceed
the number of variables. By applying tools from random matrix theory, we
characterize the asymptotic performance of such estimators when the numbers of
samples and variables grow large together. In particular, our results show
that, when outliers are absent, many estimators of the regularized-Maronna type
share the same asymptotic performance, and for these estimators we present a
data-driven method for choosing the asymptotically optimal regularization
parameter with respect to a quadratic loss. Robustness in the presence of
outliers is then studied: in the non-regularized case, a large-dimensional
robustness metric is proposed, and explicitly computed for two particular types
of estimators, exhibiting interesting differences depending on the underlying
contamination model. The impact of outliers in regularized estimators is then
studied, with interesting differences with respect to the non-regularized case,
leading to new practical insights on the choice of particular estimators.
