Marčenko-Pastur law for Tyler's M-estimator

Teng Zhang, Xiuyuan Cheng, Amit Singer

Research output: Contribution to journalArticle

4 Scopus citations

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 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.

Original languageEnglish (US)
Pages (from-to)114-123
Number of pages10
JournalJournal of Multivariate Analysis
Volume149
DOIs
StatePublished - Jul 1 2016

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Numerical Analysis
  • Statistics, Probability and Uncertainty

Keywords

  • Covariance estimation
  • Random matrix theory
  • Robust statistics

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