High dimensional limit theorems and matrix decompositions on the Stiefel manifold

The main purpose of this paper is to investigate high dimensional limiting behaviors, as m becomes infinite (m → ∞), of matrix statistics on the Stiefel manifold V_{k, m}, which consists of m × k (m ≥ k) matrices X such that X′X = I_k. The results extend those of Watson. Let X be a random matrix on V_{k, m}. We present a matrix decomposition of X as the sum of mutually orthogonal singular value decompositions of the projections P X and P X, where and are each a subspace of R^m of dimension p and their orthogonal compliment, respectively (p ≥ k and m ≥ k + p). Based on this decomposition of X, the invariant measure on V_{k, m} is expressed as the product of the measures on the component subspaces. Some distributions related to these decompositions are obtained for some population distributions on V_{k, m}. We show the limiting normalities, as m → ∞, of some matrix statistics derived from the uniform distribution and the distributions having densities of the general forms f(P X) and f(m^1/2P X) on V_{k, m}. Subsequently, applications of these high dimensional limit theorems are considered in some testing problems.     

Distributions of orientations on Stiefel manifolds

The Riemann space whose elements are m × k (m k) matrices X, i.e., orientations, such that X′X = I_k is called the Stiefel manifold V_k,m. The matrix Langevin (or von Mises-Fisher) and matrix Bingham distributions have been suggested as distributions on V_k,m. In this paper, we present some distributional results on V_k,m. Two kinds of decomposition are given of the differential form for the invariant measure on V_k,m, and they are utilized to derive distributions on the component Stiefel manifolds and subspaces of V_k,m for the above-mentioned two distributions. The singular value decomposition of the sum of a random sample from the matrix Langevin distribution gives the maximum likelihood estimators of the population orientations and modal orientation. We derive sampling distributions of matrix statistics including these sample estimators. Furthermore, representations in terms of the Hankel transform and multi-sample distribution theory are briefly discussed.     

Density Estimation on the Stiefel Manifold

This paper develops the theory of density estimation on the Stiefel manifoldV_{k, m}, whereV_{k, m}is represented by the set ofm×kmatricesXsuch thatX′X=I_k, thek×kidentity matrix. The density estimation by the method of kernels is considered, proposing two classes of kernel density estimators with small smoothing parameter matrices and for kernel functions of matrix argument. Asymptotic behavior of various statistical measures of the kernel density estimators is investigated for small smoothing parameter matrix and/or for large sample size. Some decompositions of the Stiefel manifoldV_{k, m}play useful roles in the investigation, and the general discussion is applied and examined for a special kernel function. Alternative methods of density estimation are suggested, using decompositions ofV_{k, m}.     

Expressions for some hypergeometric functions of matrix argument with applications

Reasonably simple expressions are given for some hypergeometric functions when the size of the argument matrix or matrices is two. Applications of these expressions in connection with the distributions of the latent roots of a 2 × 2 Wishart matrix are also given.     

On the evaluation of some distributions that arise in simultaneous tests for the equality of the latent roots of the covariance matrix

In this paper, the authors consider the evaluation of the distribution functions of the ratios of the intermediate roots to the trace of the real Wishart matrix as well as the ratios of the individual roots to the trace of the complex Wishart matrix. In addition, the authors consider the evaluation of the distribution functions of the ratios of the extreme roots of the Wishart matrix in the real and complex cases. Some applications and tables of the above distributions are also given.