Marčenko-Pastur Law for Tyler’s and Maronna’s M-estimators

This paper studies the limiting behavior of Tyler’s and Maronna’s Mestimators, in the regime that the number of samples n and the dimension p both go to infinity, and p/n converges to a constant y with 0 < y < 1. We prove that when the data samples are identically and independently generated from the Gaussian distribution N(0, I), the difference between 1 n ∑n i=1 xix T i and a scaled version of Tyler’s M-estimator or Maronna’s M-estimator tends to zero in spectral norm, and the empirical spectral densities of both estimators converge to the Marčenko-Pastur distribution. We also prove that when the data samples are generated from an elliptical distribution, the limiting distribution of Tyler’s M-estimator converges to a Marčenko-Pastur-Type distribution. 1 ar X iv :1 40 1. 34 24 v3 [ m at h. ST ] 1 D ec 2 01 4