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A subspace estimator for fixed rank perturbations of large random matrices

Author

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  • Hachem, Walid
  • Loubaton, Philippe
  • Mestre, Xavier
  • Najim, Jamal
  • Vallet, Pascal

Abstract

This paper deals with the problem of parameter estimation based on certain eigenspaces of the empirical covariance matrix of an observed multidimensional time series, in the case where the time series dimension and the observation window grow to infinity at the same pace. In the area of large random matrix theory, recent contributions studied the behavior of the extreme eigenvalues of a random matrix and their associated eigenspaces when this matrix is subject to a fixed-rank perturbation. The present work is concerned with the situation where the parameters to be estimated determine the eigenspace structure of a certain fixed-rank perturbation of the empirical covariance matrix. An estimation algorithm in the spirit of the well-known MUSIC algorithm for parameter estimation is developed. It relies on an approach recently developed by Benaych-Georges and Nadakuditi (2011) [8,9], relating the eigenspaces of extreme eigenvalues of the empirical covariance matrix with eigenspaces of the perturbation matrix. First and second order analyses of the new algorithm are performed.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:114:y:2013:i:c:p:427-447
    DOI: 10.1016/j.jmva.2012.08.006
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    References listed on IDEAS

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    1. Nadler, Boaz, 2011. "On the distribution of the ratio of the largest eigenvalue to the trace of a Wishart matrix," Journal of Multivariate Analysis, Elsevier, vol. 102(2), pages 363-371, February.
    2. 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.
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    Cited by:

    1. Banna, Marwa & Najim, Jamal & Yao, Jianfeng, 2020. "A CLT for linear spectral statistics of large random information-plus-noise matrices," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 2250-2281.
    2. 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.
    3. Emmanuelle Jay & Thibault Soler & Eugénie Terreaux & Jean-Philippe Ovarlez & Frédéric Pascal & Philippe De Peretti & Christophe Chorro, 2019. "Improving portfolios global performance using a cleaned and robust covariance matrix estimate," Documents de travail du Centre d'Economie de la Sorbonne 19022, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    4. Emmanuelle Jay & Thibault Soler & Eugénie Terreaux & Jean-Philippe Ovarlez & Frédéric Pascal & Philippe de Peretti & Christophe Chorro, 2019. "Improving portfolios global performance using a cleaned and robust covariance matrix estimate," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-02354596, HAL.
    5. Couillet, Romain, 2015. "Robust spiked random matrices and a robust G-MUSIC estimator," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 139-161.
    6. Emmanuelle Jay & Thibault Soler & Eugénie Terreaux & Jean-Philippe Ovarlez & Frédéric Pascal & Philippe de Peretti & Christophe Chorro, 2019. "Improving portfolios global performance using a cleaned and robust covariance matrix estimate," Post-Print halshs-02354596, HAL.

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