Inference on the eigenvalues of the covariance matrix of a multivariate normal distribution—Geometrical view

We consider inference on the eigenvalues of the covariance matrix of a multivariate normal distribution. The family of multivariate normal distributions with a fixed mean is seen as a Riemannian manifold with Fisher information metric. Two submanifolds naturally arises: one is the submanifold given by the fixed eigenvectors of the covariance matrix; the other is the one given by the fixed eigenvalues. We analyze the geometrical structures of these manifolds such as metric, embedding curvature under e-connection or m-connection. Based on these results, we study (1) the bias of the sample eigenvalues, (2) the asymptotic variance of estimators, (3) the asymptotic information loss caused by neglecting the sample eigenvectors, (4) the derivation of a new estimator that is natural from a geometrical point of view.