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Asymptotic distribution of Wishart matrix for block-wise dispersion of population eigenvalues

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  • Sheena, Yo
  • Takemura, Akimichi

Abstract

This paper deals with the asymptotic distribution of Wishart matrix and its application to the estimation of the population matrix parameter when the population eigenvalues are block-wise infinitely dispersed. We show that the appropriately normalized eigenvectors and eigenvalues asymptotically generate two Wishart matrices and one normally distributed random matrix, which are mutually independent. For a family of orthogonally equivariant estimators, we calculate the asymptotic risks with respect to the entropy or the quadratic loss function and derive the asymptotically best estimator among the family. We numerically show (1) the convergence in both the distributions and the risks are quick enough for a practical use, (2) the asymptotically best estimator is robust against the deviation of the population eigenvalues from the block-wise infinite dispersion.

Suggested Citation

  • Sheena, Yo & Takemura, Akimichi, 2008. "Asymptotic distribution of Wishart matrix for block-wise dispersion of population eigenvalues," Journal of Multivariate Analysis, Elsevier, vol. 99(4), pages 751-775, April.
    • Handle: RePEc:eee:jmvana:v:99:y:2008:i:4:p:751-775
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    References listed on IDEAS

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    1. Sheena, Yo & Takemura, Akimichi, 1992. "Inadmissibility of non-order-preserving orthogonally invariant estimators of the covariance matrix in the case of Stein's loss," Journal of Multivariate Analysis, Elsevier, vol. 41(1), pages 117-131, April.
    2. Takemura, Akimichi & Sheena, Yo, 2005. "Distribution of eigenvalues and eigenvectors of Wishart matrix when the population eigenvalues are infinitely dispersed and its application to minimax estimation of covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 94(2), pages 271-299, June.
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