Estimation of the inverse scatter matrix of an elliptically symmetric distribution

Abstract

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 a S − 1 or a S + 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 ( Σ ˆ − 1 − Σ − 1 ) 2 . To this end, a new and general Stein–Haff identity is derived for the high-dimensional elliptically symmetric distribution setting.