Expected likelihood approach for low sample support covariance matrix estimation in angular central Gaussian distributions

We address the problem of estimating the covariance matrix from a complex central angular Gaussian distribution when the number of samples T is less than the size of the observation space M. As regularization is needed, we consider the expected likelihood (EL) approach as a means to set the regularization parameters. The EL principle, originally developed under the Gaussian assumption, relies on some invariance properties of the likelihood ratio (LR). In this paper, we show that the LR, evaluated at the true covariance matrix, has a distribution that only depends on T and M. A simple representation as a product of beta distributed random variables is presented. This paves the way to EL-based regularized covariance matrix estimation, whose effectiveness is shown through simulations.