Asymptotic Distribution of Wishart Matrix for Block-wise Dispersion of Population Eigenvalues

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.