  We consider the problem of estimating high-dimensional covariance matrices of
$K$-populations or classes in the setting where the sample sizes are comparable
to the data dimension. We propose estimating each class covariance matrix as a
distinct linear combination of all class sample covariance matrices. This
approach is shown to reduce the estimation error when the sample sizes are
limited, and the true class covariance matrices share a somewhat similar
structure. We develop an effective method for estimating the coefficients in
the linear combination that minimize the mean squared error under the general
assumption that the samples are drawn from (unspecified) elliptically symmetric
distributions possessing finite fourth-order moments. To this end, we utilize
the spatial sign covariance matrix, which we show (under rather general
conditions) to be an asymptotically unbiased estimator of the normalized
covariance matrix as the dimension grows to infinity. We also show how the
proposed method can be used in choosing the regularization parameters for
multiple target matrices in a single class covariance matrix estimation
problem. We assess the proposed method via numerical simulation studies
including an application in global minimum variance portfolio optimization
using real stock data.
