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Estimation of Covariance and Precision Matrices in High Dimension

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  • Tatsuya Kubokawa

    (Faculty of Economics, University of Tokyo)

  • Akira Inoue

    (Graduate School of Economics, University of Tokyo)

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    Abstract

    The problem of estimating covariance and precision matrices of multivariate normal distributions is addressed when both the sample size and the dimension of variables are large. The estimation of the precision matrix is important in various statistical inference including the Fisher linear discriminant analysis, confidence region based on the Mahalanobis distance and others. A standard estimator is the inverse of the sample covariance matrix, but it may be instable or can not be defined in the high dimension. Although (adaptive) ridge type estimators are alternative procedures which are useful and stable for large dimension. However, we are faced with questions about how to choose ridge parameters and their estimators and how to set up asymptotic order in ridge functions in high dimensional cases. In this paper, we consider general types of ridge estimators for covariance and precision matrices, and derive asymptotic expansions of their risk functions. Then we suggest the ridge functions so that the second order terms of risks of ridge estimators are smaller than those of risks of the standard estimators.

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    File URL: http://www.cirje.e.u-tokyo.ac.jp/research/dp/2012/2012cf855.pdf
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    Bibliographic Info

    Paper provided by CIRJE, Faculty of Economics, University of Tokyo in its series CIRJE F-Series with number CIRJE-F-855.

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    Length: 30 pages
    Date of creation: Jul 2012
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    Handle: RePEc:tky:fseres:2012cf855

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    1. 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.
    2. 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.
    3. Fisher, Thomas J. & Sun, Xiaoqian, 2011. "Improved Stein-type shrinkage estimators for the high-dimensional multivariate normal covariance matrix," Computational Statistics & Data Analysis, Elsevier, vol. 55(5), pages 1909-1918, May.
    4. Jushan Bai & Shuzhong Shi, 2011. "Estimating High Dimensional Covariance Matrices and its Applications," Annals of Economics and Finance, Society for AEF, vol. 12(2), pages 199-215, November.
    5. 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.
    6. 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.
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