Large-Dimensional Behavior of Regularized Maronna's M-Estimators of Covariance Matrices - 2018


Robust estimators of large covariance matrices are thought-about, comprising regularized (linear shrinkage) modifications of Maronna's classical M-estimators. These estimators offer robustness to outliers, whereas simultaneously being well-outlined when the number of samples will not exceed the amount of variables. By applying tools from random matrix theory, we tend to characterize the asymptotic performance of such estimators when the numbers of samples and variables grow massive along. In explicit, our results show that, when outliers are absent, several estimators of the regularized-Maronna kind share the same asymptotic performance, and for these estimators, we present a data-driven methodology for choosing the asymptotically optimal regularization parameter with respect to a quadratic loss. Robustness in the presence of outliers is then studied: within the nonregularized case, a massive-dimensional robustness metric is proposed, and explicitly computed for 2 explicit varieties of estimators, exhibiting interesting differences relying on the underlying contamination model. The impact of outliers in regularized estimators is then studied, with interesting differences with respect to the nonregularized case, leading to new practical insights on the selection of explicit estimators.

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