Marcenko-Pastur law for Tyler’s M-estimator

Abstract

This paper studies the limiting behavior of Tyler’s M-estimator for the scatter matrix, in the regime that the number of samples n and their dimension p both go to infinity, and pin converges to a constant y with 0 < y < 1. We prove that when the data samples, x(1), ... , x(n) are identically and independently generated from the Gaussian distribution (0, 1), the operator norm of the difference between a properly scaled Tyler’s M-estimator and Sigma(n)(i=1)xixi T/n tends to zero. As a result, the spectral distribution of Tyler’s M-estimator converges weakly to the Marcenko-Pastur distribution. (C) 2016 Elsevier Inc. All rights reserved.