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Estimation of two high-dimensional covariance matrices and the spectrum of their ratio

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  • Wen, Jun

Abstract

Let Sp,1, Sp,2 be two independent p×p sample covariance matrices with degrees of freedom n1 and n2, respectively, whose corresponding population covariance matrices are Σp,1 and Σp,2, respectively. Knowing Sp,1, Sp,2, this article proposes a class of estimators for the spectrum (eigenvalues) of the matrix Σp,2Σp,1−1 as well as the pair of the whole matrices (Σp,1,Σp,2). The estimators are created based on Random Matrix Theory. Under mild conditions, our estimator for the spectrum of Σp,2Σp,1−1 is shown to be weakly consistent and the estimator for (Σp,1,Σp,2) is shown to be optimal in the sense of minimizing the asymptotic loss within the class of equivariant estimators as n1,n2,p→∞ with p∕n1→c1∈(0,1), p∕n2→c2∈(0,1)∪(1,∞). Also, our estimators are easy to implement. Even when p is 1000, our estimators can be computed in seconds using a personallaptop.

Suggested Citation

  • Wen, Jun, 2018. "Estimation of two high-dimensional covariance matrices and the spectrum of their ratio," Journal of Multivariate Analysis, Elsevier, vol. 168(C), pages 1-29.
    • Handle: RePEc:eee:jmvana:v:168:y:2018:i:c:p:1-29
    DOI: 10.1016/j.jmva.2018.06.008
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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. Pui Leung & Wai Chan, 1998. "Estimation of the Scale Matrix and its Eigenvalues in the Wishart and the Multivariate F Distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 50(3), pages 523-530, September.
    4. Michael J. Daniels & Robert E. Kass, 2001. "Shrinkage Estimators for Covariance Matrices," Biometrics, The International Biometric Society, vol. 57(4), pages 1173-1184, December.
    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. Yoshihiko Konno, 1991. "A note on estimating eigenvalues of scale matrix of the multivariate F-distribution," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 43(1), pages 157-165, March.
    7. Ledoit, Olivier & Wolf, Michael, 2015. "Spectrum estimation: A unified framework for covariance matrix estimation and PCA in large dimensions," Journal of Multivariate Analysis, Elsevier, vol. 139(C), pages 360-384.
    8. Weiming Li & Jianfeng Yao, 2015. "On generalized expectation-based estimation of a population spectral distribution from high-dimensional data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(2), pages 359-373, April.
    9. 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.
    10. Loh, Wei-Liem, 1991. "Estimating covariance matrices II," Journal of Multivariate Analysis, Elsevier, vol. 36(2), pages 163-174, February.
    11. Dipak Dey, 1988. "Simultaneous estimation of eigenvalues," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 40(1), pages 137-147, March.
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