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The random matrix regime of Maronna’s M-estimator with elliptically distributed samples

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  • Couillet, Romain
  • Pascal, Frédéric
  • Silverstein, Jack W.

Abstract

This article demonstrates that the robust scatter matrix estimator CˆN∈CN×N of a multivariate elliptical population x1,…,xn∈CN originally proposed by Maronna in 1976, and defined as the solution (when existent) of an implicit equation, behaves similar to a well-known random matrix model in the limiting regime where the population N and sample n sizes grow at the same speed. We show precisely that CˆN∈CN×N is defined for all n large with probability one and that, under some light hypotheses, ‖CˆN−SˆN‖→0 almost surely in spectral norm, where SˆN follows a classical random matrix model. As a corollary, the limiting eigenvalue distribution of CˆN is derived. This analysis finds applications in the fields of statistical inference and signal processing.

Suggested Citation

  • Couillet, Romain & Pascal, Frédéric & Silverstein, Jack W., 2015. "The random matrix regime of Maronna’s M-estimator with elliptically distributed samples," Journal of Multivariate Analysis, Elsevier, vol. 139(C), pages 56-78.
  • Handle: RePEc:eee:jmvana:v:139:y:2015:i:c:p:56-78
    DOI: 10.1016/j.jmva.2015.02.020
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    References listed on IDEAS

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    1. Couillet, Romain & McKay, Matthew, 2014. "Large dimensional analysis and optimization of robust shrinkage covariance matrix estimators," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 99-120.
    2. Hachem, Walid & Loubaton, Philippe & Mestre, Xavier & Najim, Jamal & Vallet, Pascal, 2013. "A subspace estimator for fixed rank perturbations of large random matrices," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 427-447.
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    4. Silverstein, J. W., 1995. "Strong Convergence of the Empirical Distribution of Eigenvalues of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 55(2), pages 331-339, November.
    5. Silverstein, J. W. & Choi, S. I., 1995. "Analysis of the Limiting Spectral Distribution of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 54(2), pages 295-309, August.
    6. Baik, Jinho & Silverstein, Jack W., 2006. "Eigenvalues of large sample covariance matrices of spiked population models," Journal of Multivariate Analysis, Elsevier, vol. 97(6), pages 1382-1408, July.
    7. Paul, Debashis & Silverstein, Jack W., 2009. "No eigenvalues outside the support of the limiting empirical spectral distribution of a separable covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 100(1), pages 37-57, January.
    8. Silverstein, J. W. & Bai, Z. D., 1995. "On the Empirical Distribution of Eigenvalues of a Class of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 54(2), pages 175-192, August.
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    Cited by:

    1. Ciobotaru, Corina & Mazza, Christian, 2022. "Consistency and asymptotic normality of M-estimates of scatter on Grassmann manifolds," Journal of Multivariate Analysis, Elsevier, vol. 190(C).
    2. 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.
    3. Couillet, Romain, 2015. "Robust spiked random matrices and a robust G-MUSIC estimator," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 139-161.
    4. Joel Bun & Jean-Philippe Bouchaud & Marc Potters, 2016. "Cleaning large correlation matrices: tools from random matrix theory," Papers 1610.08104, arXiv.org.
    5. Ding, Hao & Qin, Shanshan & Wu, Yuehua & Wu, Yaohua, 2021. "Asymptotic properties on high-dimensional multivariate regression M-estimation," Journal of Multivariate Analysis, Elsevier, vol. 183(C).
    6. Romanov, Elad & Kur, Gil & Nadler, Boaz, 2023. "Tyler’s and Maronna’s M-estimators: Non-asymptotic concentration results," Journal of Multivariate Analysis, Elsevier, vol. 196(C).
    7. Couillet, Romain & Kammoun, Abla & Pascal, Frédéric, 2016. "Second order statistics of robust estimators of scatter. Application to GLRT detection for elliptical signals," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 249-274.

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