On the Convergence of Maronna's M-Estimators of Scatter

Yacine Chitour (1) , Romain Couillet (1) , Frédéric Pascal (2)

1 L2S - Laboratoire des signaux et systèmes
2 SONDRA - Sondra, CentraleSupélec, Université Paris-Saclay (COmUE)

In this letter, we propose an alternative proof for the uniqueness of Maronna's M-estimator of scatter [1] for vector observations under a mild constraint of linear independence of any subset of of these vectors. This entails in particular almost sure uniqueness for random vectors with a density as long as. This approach allows to establish further relations that demonstrate that a properly normalized Tyler's-estimator of scatter [2] can be considered as a limit of Maronna's M-estimator. More precisely, the contribution is to show that each-estimator, verifying some mild conditions, converges towards a particular Tyler's-estimator. These results find important implications in recent works on the large dimensional (random matrix) regime of robust-estimation.

Keywords: Covariance matrix estimation; Tyler's estimator; M-estimators

Domains: Signal and Image Processing; Other Statistics [stat.ML]; Applications [stat.AP]

Yacine Chitour, Romain Couillet, Frédéric Pascal. On the Convergence of Maronna's M-Estimators of Scatter. IEEE Signal Processing Letters, 2015, 22 (6), pp.709-712. ⟨10.1109/LSP.2014.2367547⟩. ⟨hal-01251753⟩