Robust shrinkage estimation of high-dimensional covariance matrices

We address high dimensional covariance estimation for elliptical distributed samples, which are also known as spherically invariant random vectors (SIRV) or compound-Gaussian processes. Specifically we consider shrinkage methods that are suitable for high dimensional problems with a small number of samples (large p small n). We start from a classical robust covariance estimator [Tyler (1987)], which is distribution-free within the family of elliptical distribution but inapplicable when n<p. Using a shrinkage coefficient, we regularize Tyler's fixed-point iterations. We prove that, for all n and p, the proposed fixed-point iterations converge to a unique limit regardless of the initial condition. Next, we propose a simple, closed-form and data dependent choice for the shrinkage coefficient, which is based on a minimum mean squared error framework. Simulations demonstrate that the proposed method achieves low estimation error and is robust to heavy-tailed samples. Finally, as a real-world application we demonstrate the performance of the proposed technique in the context of activity/intrusion detection using a wireless sensor network.