An asymptotic expansion of Wishart distribution when the population eigenvalues are infinitely dispersed

Takemura and Sheena (2002) derived the asymptotic joint distribution of the eigenvalues and the eigenvectors of Wishart matrix when the population eigenvalues become infinitely dispersed. They also showed necessary conditions for an estimator of the population covariance matrix to be minimax for typical loss functions by calculating the asymptotic risk of the estimator. In this paper, we further examine those distributions and risks by means of an asymptotic expansion. We focus on a limiting process where the population eigenvalues become linearly dispersed, which can be parametrized by one parameter. We obtain the asymptotic expansion of the distribution function of relevant elements of the sample eigenvalues and eigenvectors with respect to the parameter. We also derive the asymptotic expansion of the risk function of a scale and orthogonally equivariant estimator. As an application, we prove non-minimaxity of Stein’s and Haff’s estimators, which has been an open problem so far.