High dimensional asymptotic expansions for the matrix Langevin distributions on the Stiefel manifold
type: Item
instance of: scholarly article
title: High dimensional asymptotic expansions for the matrix Langevin distributions on the Stiefel manifold
zbMATH Open document ID: 0776.62048
author: Yasuko Chikuse
published in: Journal of Multivariate Analysis
publication date: 1993-05-16T00:00:00Z
review text: From geometrical point of view, a high dimensional Stiefel manifold, say \(V_{k,m}\), defined as a set of \(m\times k\) real matrices whose \(m\)- variate column vectors are mutually orthogonal, can be used for analyzing and solving some sophisticated statistics problems. For example, it can be applied to analyze compositional data with an arbitrary number \(m\) of components. As main results of this paper, two versions of the matrix Langevin distribution based on the Stiefel manifold are investigated. Furthermore, for the two distribution versions and large \(m\), the asymptotic expansions for the probability density functions and for some matrix variables and their related scale variables are derived, respectively, in terms of invariant zonal polynomials. In addition, some applications of these asymptotic expansions are provided, such as hypothesis testing problems on Stiefel manifolds, which show that the asymptotic expansions derived in this paper are useful for statistical inference on Stiefel manifolds.
Mathematics Subject Classification ID: 62H10
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 62H11
Mathematics Subject Classification ID: 58C35
zbMATH DE Number: 168195
zbMATH Keywords: singular value decomposition
zbMATH Keywords: random matrix
zbMATH Keywords: matrix uniform and Langevin distributions
zbMATH Keywords: matrix-variate normal distributions
zbMATH Keywords: Wishart distributions
zbMATH Keywords: Hermite and Laguerre polynomials
zbMATH Keywords: Rodrigues formulae
zbMATH Keywords: polynomials of matrix arguments
zbMATH Keywords: high dimensional Stiefel manifold
zbMATH Keywords: matrix Langevin distribution
zbMATH Keywords: asymptotic expansions
zbMATH Keywords: density functions
zbMATH Keywords: matrix variables
zbMATH Keywords: scale variables
zbMATH Keywords: invariant zonal polynomials
zbMATH Keywords: hypothesis testing problems
reviewed by: Kai-Tai Fang
MaRDI profile type: Publication
full work available at URL: jmva.1993.1005
OpenAlex ID: W1987332671
DOI: 10.1006/JMVA.1993.1005
description: scientific article; zbMATH DE number 168195