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Estimation of a high-dimensional covariance matrix with the Stein loss

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  • Tsukuma, Hisayuki

Abstract

The problem of estimating a normal covariance matrix is considered from a decision-theoretic point of view, where the dimension of the covariance matrix is larger than the sample size. This paper addresses not only the nonsingular case but also the singular case in terms of the covariance matrix. Based on James and Stein’s minimax estimator and on an orthogonally invariant estimator, some classes of estimators are unifiedly defined for any possible ordering on the dimension, the sample size and the rank of the covariance matrix. Unified dominance results on such classes are provided under a Stein-type entropy loss. The unified dominance results are applied to improving on an empirical Bayes estimator of a high-dimensional covariance matrix.

Suggested Citation

  • Tsukuma, Hisayuki, 2016. "Estimation of a high-dimensional covariance matrix with the Stein loss," Journal of Multivariate Analysis, Elsevier, vol. 148(C), pages 1-17.
    • Handle: RePEc:eee:jmvana:v:148:y:2016:i:c:p:1-17
    DOI: 10.1016/j.jmva.2016.02.012
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    References listed on IDEAS

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    8. Bodnar, Taras & Mazur, Stepan & Podgórski, Krzysztof, 2016. "Singular inverse Wishart distribution and its application to portfolio theory," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 314-326.
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    Cited by:

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    2. Haddouche, Anis M. & Fourdrinier, Dominique & Mezoued, Fatiha, 2021. "Scale matrix estimation of an elliptically symmetric distribution in high and low dimensions," Journal of Multivariate Analysis, Elsevier, vol. 181(C).
    3. Alfelt, Gustav & Mazur, Stepan, 2020. "On the mean and variance of the estimated tangency portfolio weights for small samples," Working Papers 2020:8, Örebro University, School of Business.
    4. Karlsson, Sune & Mazur, Stepan & Muhinyuza, Stanislas, 2020. "Statistical Inference for the Tangency Portfolio in High Dimension," Working Papers 2020:10, Örebro University, School of Business.
    5. Besson, Olivier & Vincent, François & Gendre, Xavier, 2020. "A Stein’s approach to covariance matrix estimation using regularization of Cholesky factor and log-Cholesky metric," Statistics & Probability Letters, Elsevier, vol. 167(C).
    6. Fourdrinier, Dominique & Haddouche, Anis M. & Mezoued, Fatiha, 2021. "Covariance matrix estimation under data-based loss," Statistics & Probability Letters, Elsevier, vol. 177(C).

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