  This paper considers the problem of estimating a high-dimensional (HD)
covariance matrix when the sample size is smaller, or not much larger, than the
dimensionality of the data, which could potentially be very large. We develop a
regularized sample covariance matrix (RSCM) estimator which can be applied in
commonly occurring sparse data problems. The proposed RSCM estimator is based
on estimators of the unknown optimal (oracle) shrinkage parameters that yield
the minimum mean squared error (MMSE) between the RSCM and the true covariance
matrix when the data is sampled from an unspecified elliptically symmetric
distribution. We propose two variants of the RSCM estimator which differ in the
approach in which they estimate the underlying sphericity parameter involved in
the theoretical optimal shrinkage parameter. The performance of the proposed
RSCM estimators are evaluated with numerical simulation studies. In particular
when the sample sizes are low, the proposed RSCM estimators often show a
significant improvement over the conventional RSCM estimator by Ledoit and Wolf
(2004). We further evaluate the performance of the proposed estimators in
classification and portfolio optimization problems with real data wherein the
proposed methods are able to outperform the benchmark methods.
