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

Publication Type:

Journal Article

Source:

IEEE Signal Processing Letters, Volume 22, Issue 6, p.709-712 (2014)

Abstract:

In this paper, we propose an alternative proof for the uniqueness of Maronna’s M-estimator of scatter [1] for N vector observations y1, . . . , yN ∈ Rm 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 yi with a density as long as N > m. This approach allows to establish further relations that demonstrate that a properly normalized Tyler’s Mestimator of scatter [2] can be considered as a limit of Maronna’s M-estimator. More precisely, the contribution is to show that each M-estimator, verifying some mild conditions, 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.