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Second order statistics of robust estimators of scatter. Application to GLRT detection for elliptical signals

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  • Couillet, Romain
  • Kammoun, Abla
  • Pascal, Frédéric

Abstract

A central limit theorem for bilinear forms of the type a∗CˆN(ρ)−1b, where a,b∈CN are unit norm deterministic vectors and CˆN(ρ) a robust-shrinkage estimator of scatter parametrized by ρ and built upon n independent elliptical vector observations, is presented. The fluctuations of a∗CˆN(ρ)−1b are found to be of order N−12 and to be the same as those of a∗SˆN(ρ)−1b for SˆN(ρ) a matrix of a theoretical tractable form. This result is exploited in a classical signal detection problem to provide an improved detector which is both robust to elliptical data observations (e.g., impulsive noise) and optimized across the shrinkage parameter ρ.

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  • 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.
    • Handle: RePEc:eee:jmvana:v:143:y:2016:i:c:p:249-274
    DOI: 10.1016/j.jmva.2015.08.021
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    References listed on IDEAS

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    1. Ledoit, Olivier & Wolf, Michael, 2004. "A well-conditioned estimator for large-dimensional covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 88(2), pages 365-411, February.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. Couillet, Romain, 2015. "Robust spiked random matrices and a robust G-MUSIC estimator," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 139-161.
    7. 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. 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. Joel Bun & Jean-Philippe Bouchaud & Marc Potters, 2016. "Cleaning large correlation matrices: tools from random matrix theory," Papers 1610.08104, arXiv.org.
    3. 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).

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