On the Expected Likelihood Approach for Assessment of Regularization Covariance Matrix

Regularization, which consists in shrinkage of the sample covariance matrix to a target matrix, is a commonly used and effective technique in low sample support covariance matrix estimation. Usually, a target matrix is chosen and optimization of the shrinkage factor is carried out, based on some relevant metric. In this letter, we rather address the choice of the target matrix. More precisely, we aim at evaluating, from observation of the data matrix, whether a given target matrix is a good regularizer. Towards this end, the expected likelihood (EL) approach is investigated. At a first step, we re-interpret the regularized covariance matrix estimate as the minimum mean-square error estimate in a Bayesian model where the target matrix serves as a prior. The likelihood function of the data is then derived, and the EL principle is subsequently applied. Over-sampled and under-sampled scenarios are considered.