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Large dimensional analysis and optimization of robust shrinkage covariance matrix estimators

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  • Couillet, Romain
  • McKay, Matthew

Abstract

This article studies two regularized robust estimators of scatter matrices proposed (and proved to be well defined) in parallel in Chen et al. (2011) and Pascal et al. (2013), based on Tyler’s robust M-estimator (Tyler, 1987) and on Ledoit and Wolf’s shrinkage covariance matrix estimator (Ledoit and Wolf, 2004). These hybrid estimators have the advantage of conveying (i) robustness to outliers or impulsive samples and (ii) small sample size adequacy to the classical sample covariance matrix estimator. We consider here the case of i.i.d. elliptical zero mean samples in the regime where both sample and population sizes are large. We demonstrate that, under this setting, the estimators under study asymptotically behave similar to well-understood random matrix models. This characterization allows us to derive optimal shrinkage strategies to estimate the population scatter matrix, improving significantly upon the empirical shrinkage method proposed in Chen et al. (2011).

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  • 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.
    • Handle: RePEc:eee:jmvana:v:131:y:2014:i:c:p:99-120
    DOI: 10.1016/j.jmva.2014.06.018
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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. 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.
    3. Ledoit, Olivier & Wolf, Michael, 2003. "Improved estimation of the covariance matrix of stock returns with an application to portfolio selection," Journal of Empirical Finance, Elsevier, vol. 10(5), pages 603-621, December.
    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. Schäfer Juliane & Strimmer Korbinian, 2005. "A Shrinkage Approach to Large-Scale Covariance Matrix Estimation and Implications for Functional Genomics," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 4(1), pages 1-32, November.
    6. 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. Liusha Yang & Romain Couillet & Matthew R. McKay, 2015. "A Robust Statistics Approach to Minimum Variance Portfolio Optimization," Papers 1503.08013, arXiv.org.
    2. Couillet, Romain, 2015. "Robust spiked random matrices and a robust G-MUSIC estimator," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 139-161.
    3. Yuanrong Wang & Tomaso Aste, 2022. "Sparsification and Filtering for Spatial-temporal GNN in Multivariate Time-series," Papers 2203.03991, arXiv.org.
    4. 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).
    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. Jan Kalina & Jan Tichavský, 2022. "The minimum weighted covariance determinant estimator for high-dimensional data," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 16(4), pages 977-999, December.
    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.
    8. Liusha Yang & Matthew R. Mckay & Romain Couillet, 2018. "High-Dimensional MVDR Beamforming: Optimized Solutions Based on Spiked Random Matrix Models," Post-Print hal-01957672, HAL.
    9. 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).
    10. Elisa Cabana & Rosa E. Lillo & Henry Laniado, 2021. "Multivariate outlier detection based on a robust Mahalanobis distance with shrinkage estimators," Statistical Papers, Springer, vol. 62(4), pages 1583-1609, August.
    11. Benoit Oriol & Alexandre Miot, 2023. "Ledoit-Wolf linear shrinkage with unknown mean," Papers 2304.07045, arXiv.org.

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