Asymptotic distribution of Wishart matrix for block-wise dispersion of population eigenvalues |
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Authors: | Yo Sheena Akimichi Takemura |
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Affiliation: | a Department of Economics, Shinshu University, 3-1-1 Asahi, Matsumoto, Nagano 3908621, Japan b Graduate School of Information Science and Technology, University of Tokyo, Japan |
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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. |
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Keywords: | 62H10 62C20 |
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