Semicircle law of Tyler’s M-estimator for scatter

This paper analyzes the spectral properties of Tyler’s M-estimator for scatter Tn,d. It is shown that if a multivariate sample stems from a generalized spherically distributed population and the sample size n and the dimension d both go to infinity while d/n→0, then the empirical spectral distribution of n/d(Tn,d−Id), Id being the identity, converges in probability to the semicircle law. In contrast to that of the sample covariance matrix, this convergence does not necessarily require the sample vectors to be componentwise independent. Further, moments of the generalized spherical population do not have to exist.