  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.
