On the convergence of Maronna's $M$-estimators of scatter

Abstract

In this paper, {we propose an alternative proof for the uniqueness} of Maronna's $M$-estimator of scatter (Maronna, 1976) for $N$ vector observations ${\mathbf{y}}_1,...,{\mathbf{y}}_N\in {\mathbb{R}}^m$ under a mild constraint of linear independence of any subset of $m$ of these vectors. This entails in particular almost sure uniqueness for random vectors ${\mathbf{y}}_i$ with a density as long as $N>m$. {This approach allows to establish further relations that demonstrate that a properly normalized Tyler's $M$-estimator of scatter (Tyler, 1987) can be considered as a limit of Maronna's $M$-estimator. More precisely, the contribution is to show that each $M$-estimator converges towards a particular Tyler's $M$-estimator.} These results find important implications in recent works on the large dimensional (random matrix) regime of robust $M$-estimation.

Cited by in corpus (1)

• Large Dimensional Analysis of Robust M-Estimators of Covariance with Outliers