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 p/n converges to a constant y with 0<y<1. We prove that when the data samples x1, . . ., xn are identically and independently generated from the Gaussian distribution N(0,I), the operator norm of the difference between a properly scaled Tyler's M-estimator and ∑i=1nxixi⊤/n tends to zero. As a result, the spectral distribution of Tyler's M-estimator converges weakly to the Marčenko-Pastur distribution.
All Science Journal Classification (ASJC) codes
- Statistics and Probability
- Numerical Analysis
- Statistics, Probability and Uncertainty
- Covariance estimation
- Random matrix theory
- Robust statistics