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Semicircle law of Tyler’s M-estimator for scatter

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  • Frahm, Gabriel
  • Glombek, Konstantin

Abstract

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.

Suggested Citation

  • Frahm, Gabriel & Glombek, Konstantin, 2012. "Semicircle law of Tyler’s M-estimator for scatter," Statistics & Probability Letters, Elsevier, vol. 82(5), pages 959-964.
    • Handle: RePEc:eee:stapro:v:82:y:2012:i:5:p:959-964
    DOI: 10.1016/j.spl.2012.01.017
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    References listed on IDEAS

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    1. Frahm, Gabriel, 2009. "Asymptotic distributions of robust shape matrices and scales," Journal of Multivariate Analysis, Elsevier, vol. 100(7), pages 1329-1337, August.
    2. Frahm, Gabriel & Jaekel, Uwe, 2009. "A generalization of Tyler's M-estimators to the case of incomplete data," Discussion Papers in Econometrics and Statistics 3/07, University of Cologne, Institute of Econometrics and Statistics.
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    Citations

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    Cited by:

    1. Zhang, Teng & Cheng, Xiuyuan & Singer, Amit, 2016. "Marčenko–Pastur law for Tyler’s M-estimator," Journal of Multivariate Analysis, Elsevier, vol. 149(C), pages 114-123.
    2. Frahm, Gabriel & Nordhausen, Klaus & Oja, Hannu, 2020. "M-estimation with incomplete and dependent multivariate data," Journal of Multivariate Analysis, Elsevier, vol. 176(C).

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