Covariance matrix estimation in complex elliptic distributions using the expected likelihood approach

We consider the problem of estimating the scatter matrix in complex elliptically symmetric (CES) distributions using the expected likelihood (EL) approach. The latter, originally derived in the Gaussian case, is based on the fact that the probability density function (p.d.f.) of the likelihood ratio (LR) for the (unknown) actual covariance matrix does not depend on this matrix, and is fully specified by the matrix dimension M and the number of independent training samples T. We extend this result to CES distributions as well as to angular central Gaussian (ACG) distributions. More precisely, we prove that for CES distributions, the p.d.f. of the LR, evaluated at the true scatter matrix Σ₀, does not depend on the latter but depends on the density generator of the CES distribution. As for the ACG case, we demonstrate that the LR for Σ₀ is distribution-free. This invariance property paves the way to derivation of regularized covariance matrix estimates, where the regularization parameters are chosen from the EL principle. The relevance of such a choice for the regularization parameters is illustrated on an example with fixedpoint diagonally loaded estimates.