M-estimators of scatter with eigenvalue shrinkage
This work provides an incremental improvement in robust statistics for researchers dealing with non-Gaussian data.
The paper tackled the problem of robust covariance estimation by proposing shrinkage M-estimators with an automatic tuning method, showing that they perform comparably to standard shrinkage estimators for Gaussian data and significantly better for heavy-tailed distributions.
A popular regularized (shrinkage) covariance estimator is the shrinkage sample covariance matrix (SCM) which shares the same set of eigenvectors as the SCM but shrinks its eigenvalues toward its grand mean. 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, 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 a heavy-tailed distribution.