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Estimation of the precision matrix of a singular Wishart distribution and its application in high-dimensional data

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  • Kubokawa, Tatsuya
  • Srivastava, Muni S.

Abstract

In this article, the Stein-Haff identity is established for a singular Wishart distribution with a positive definite mean matrix but with the dimension larger than the degrees of freedom. This identity is then used to obtain estimators of the precision matrix improving on the estimator based on the Moore-Penrose inverse of the Wishart matrix under the Efron-Morris loss function and its variants. Ridge-type empirical Bayes estimators of the precision matrix are also given and their dominance properties over the usual one are shown using this identity. Finally, these precision estimators are used in a quadratic discriminant rule, and it is shown through simulation that discriminant methods based on the ridge-type empirical Bayes estimators provide higher correct classification rates.

Suggested Citation

  • Kubokawa, Tatsuya & Srivastava, Muni S., 2008. "Estimation of the precision matrix of a singular Wishart distribution and its application in high-dimensional data," Journal of Multivariate Analysis, Elsevier, vol. 99(9), pages 1906-1928, October.
    • Handle: RePEc:eee:jmvana:v:99:y:2008:i:9:p:1906-1928
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    References listed on IDEAS

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    1. Haff, L. R., 1979. "An identity for the Wishart distribution with applications," Journal of Multivariate Analysis, Elsevier, vol. 9(4), pages 531-544, December.
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    Cited by:

    1. Bodnar, Olha & Bodnar, Taras & Parolya, Nestor, 2022. "Recent advances in shrinkage-based high-dimensional inference," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    2. Chételat, Didier & Wells, Martin T., 2016. "Improved second order estimation in the singular multivariate normal model," Journal of Multivariate Analysis, Elsevier, vol. 147(C), pages 1-19.
    3. Tatsuya Kubokawa & Akira Inoue, 2012. "Estimation of Covariance and Precision Matrices in High Dimension," CIRJE F-Series CIRJE-F-855, CIRJE, Faculty of Economics, University of Tokyo.
    4. Fourdrinier, Dominique & Mezoued, Fatiha & Wells, Martin T., 2016. "Estimation of the inverse scatter matrix of an elliptically symmetric distribution," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 32-55.
    5. Bodnar, Taras & Gupta, Arjun K. & Parolya, Nestor, 2016. "Direct shrinkage estimation of large dimensional precision matrix," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 223-236.
    6. Bodnar, Taras & Dette, Holger & Parolya, Nestor, 2016. "Spectral analysis of the Moore–Penrose inverse of a large dimensional sample covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 148(C), pages 160-172.
    7. 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.
    8. Markku Kuismin & Mikko J Sillanpää, 2016. "Use of Wishart Prior and Simple Extensions for Sparse Precision Matrix Estimation," PLOS ONE, Public Library of Science, vol. 11(2), pages 1-20, February.
    9. Kourtis, Apostolos & Dotsis, George & Markellos, Raphael N., 2012. "Parameter uncertainty in portfolio selection: Shrinking the inverse covariance matrix," Journal of Banking & Finance, Elsevier, vol. 36(9), pages 2522-2531.
    10. Takayuki Yamada & Tetsuto Himeno, 2019. "Estimation of multivariate 3rd moment for high-dimensional data and its application for testing multivariate normality," Computational Statistics, Springer, vol. 34(2), pages 911-941, June.
    11. Konno, Yoshihiko, 2009. "Shrinkage estimators for large covariance matrices in multivariate real and complex normal distributions under an invariant quadratic loss," Journal of Multivariate Analysis, Elsevier, vol. 100(10), pages 2237-2253, November.
    12. Katayama, Shota & Imori, Shinpei, 2014. "Lasso penalized model selection criteria for high-dimensional multivariate linear regression analysis," Journal of Multivariate Analysis, Elsevier, vol. 132(C), pages 138-150.
    13. Tsukuma, Hisayuki, 2014. "Bayesian estimation of a bounded precision matrix," Journal of Multivariate Analysis, Elsevier, vol. 127(C), pages 160-172.
    14. Tatsuya Kubokawa & Masashi Hyodo & Muni S. Srivastava, 2011. "Asymptotic Expansion and Estimation of EPMC for Linear Classification Rules in High Dimension," CIRJE F-Series CIRJE-F-818, CIRJE, Faculty of Economics, University of Tokyo.
    15. Park, Hoyoung & Baek, Seungchul & Park, Junyong, 2022. "High-dimensional linear discriminant analysis using nonparametric methods," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    16. Ruili Sun & Tiefeng Ma & Shuangzhe Liu, 2018. "A Stein-type shrinkage estimator of the covariance matrix for portfolio selections," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 81(8), pages 931-952, November.
    17. 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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