  A large dimensional characterization of robust M-estimators of covariance (or
scatter) is provided under the assumption that the dataset comprises
independent (essentially Gaussian) legitimate samples as well as arbitrary
deterministic samples, referred to as outliers. Building upon recent random
matrix advances in the area of robust statistics, we specifically show that the
so-called Maronna M-estimator of scatter asymptotically behaves similar to
well-known random matrices when the population and sample sizes grow together
to infinity. The introduction of outliers leads the robust estimator to behave
asymptotically as the weighted sum of the sample outer products, with a
constant weight for all legitimate samples and different weights for the
outliers. A fine analysis of this structure reveals importantly that the
propensity of the M-estimator to attenuate (or enhance) the impact of outliers
is mostly dictated by the alignment of the outliers with the inverse population
covariance matrix of the legitimate samples. Thus, robust M-estimators can
bring substantial benefits over more simplistic estimators such as the
per-sample normalized version of the sample covariance matrix, which is not
capable of differentiating the outlying samples. The analysis shows that,
within the class of Maronna's estimators of scatter, the Huber estimator is
most favorable for rejecting outliers. On the contrary, estimators more similar
to Tyler's scale invariant estimator (often preferred in the literature) run
the risk of inadvertently enhancing some outliers.
