Covariance estimation in elliptical models with convex structure

We develop the General Method of Moments (GMM) Approach for estimating the covariance matrices of non-Gaussian distributions with convex structure. The GMM turns out to be a non-convex optimization problem, thus making the addition of prior knowledge in form of convex structure constraints cumbersome. We propose a different approach to this estimator and show that the Tyler's estimator can be obtained as a solution of a convexly relaxed GMM problem, thus making the imposition of convex constraints easier. This new framework provides consistent solutions which outperform the standard projection methods. As an application of this method we consider Gaussian Compound samples with Toeplitz and banded covariance matrices. We provide synthetic numerical data and demonstrate the performance advantages of our method.