Shrinking the Eigenvalues of M-Estimators of Covariance Matrix

A highly popular regularized (shrinkage) covariance matrix estimator is the shrinkage sample covariance matrix (SCM) which shares the same set of eigenvectors as the SCM but shrinks its eigenvalues toward the grand mean of the eigenvalues of the SCM. In this paper, a more general approach is considered in which the SCM is replaced by an M-estimator of scatter matrix and a fully automatic data adaptive method to compute the optimal shrinkage parameter with minimum mean squared error is proposed. Our approach permits the use of any weight function such as Gaussian, Huber's, Tyler's, or t weight functions, all of which are commonly used in M-estimation framework. Our simulation examples illustrate that shrinkage M-estimators based on the proposed optimal tuning combined with robust weight function do not loose in performance to shrinkage SCM estimator when the data is Gaussian, but provide significantly improved performance when the data is sampled from an unspecified heavy-tailed elliptically symmetric distribution. Also, real-world and synthetic stock market data validate the performance of the proposed method in practical applications.

Automated summary

The paper presents a more general approach by replacing SCM with an M-estimator of scatter matrix and proposes a fully automatic data adaptive method for computing the optimal shrinkage parameter. The simulation examples show that the proposed method outperforms SCM estimator in Gaussian data and significantly improves performance in heavy-tailed elliptically symmetric distribution data. Real-world and synthetic stock market data also validate the performance of the proposed method in practical applications.