Robust shrinkage M-estimators of large covariance matrices

Robust high dimensional covariance estimators are considered, comprising regularized (linear shrinkage) modifications of Maronna's classical M-estimators. Such estimators aim to provide robustness to outliers, while simultaneously giving well-defined solutions under high dimensional scenarios where 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 number of samples and variables grow large together. In particular, our results show that, when outliers are absent, many estimators of the shrinkage-Maronna type share the same asymptotic performance, and for such estimators we present a data-driven method for choosing the asymptotically optimal shrinkage parameter. Although our results assume an outlier-free scenario, simulations suggest that certain estimators perform substantially better than others when subjected to outlier samples.