New insights into the statistical properties of $M$-estimators

Abstract: This paper proposes an original approach to better understanding the behavior of robust scatter matrix $M$-estimators. Scatter matrices are of particular interest for many signal processing applications since the resulting performance strongly relies on the quality of the matrix estimation. In this context, $M$-estimators appear as very interesting candidates, mainly due to their flexibility to the statistical model and their robustness to outliers and/or missing data. However, the behavior of such estimators still remains unclear and not well understood since they are described by fixed-point equations that make their statistical analysis very difficult. To fill this gap, the main contribution of this work is to prove that these estimators distribution is more accurately described by a Wishart distribution than by the classical asymptotical Gaussian approximation. To that end, we propose a new `Gaussian-core' representation for Complex Elliptically Symmetric (CES) distributions and we analyze the proximity between $M$-estimators and a Gaussian-based Sample Covariance Matrix (SCM), unobservable in practice and playing only a theoretical role. To confirm our claims we also provide results for a widely used function of $M$-estimators, the Mahalanobis distance. Finally, Monte Carlo simulations for various scenarios are presented to validate theoretical results.