Estimation of the inverse scatter matrix of an elliptically symmetric distribution

We consider estimation of the inverse scatter matrices Σ−1 for high-dimensional elliptically symmetric distributions. In high-dimensional settings the sample covariance matrix S may be singular. Depending on the singularity of S, natural estimators of Σ−1 are of the form aS−1 or aS+ where a is a positive constant and S−1 and S+ are, respectively, the inverse and the Moore-Penrose inverse of S. We propose a unified estimation approach for these two cases and provide improved estimators under the quadratic loss tr(Σˆ−1−Σ−1)2. To this end, a new and general Stein-Haff identity is derived for the high-dimensional elliptically symmetric distribution setting.