The matrix angular central Gaussian distribution

Yasuko Chikuse

Journal of Multivariate Analysis, 1990, vol. 33, issue 2, 265-274

Abstract: The Riemann space whose elements are m - k (m >= k) matrices X such that X'X = Ik is called the Stiefel manifold and denoted by Vk,m. Some distributions on Vk,m, e.g., the matrix Langevin (or von Mises-Fisher) and Bingham distributions and the uniform distribution, have been defined and discussed in the literature. In this paper, we present methods to construct new kinds of distributions on Vk,m and discuss some properties of these distributions. We investigate distributions of the "orientation" HZ = Z(Z'Z)-1/2 ([epsilon]Vk,m) of an m - k random matrix Z. The general integral form of the density of HZ reduces to a simple mathematical form, when Z has the matrix-variate central normal distribution with parameter [Sigma], an m - m positive definite matrix. We may call this distribution the matrix angular central Gaussian distribution with parameter [Sigma], denoted by the MACG ([Sigma]) distribution. The MACG distribution reduces to the angular central Gaussian distribution on the hypersphere for k = 1, which has been already known. Then, we are concerned with distributions of the orientation HY of a linear transformation Y = BZ of Z, where B is an m - m matrix such that [short parallel]B[short parallel] [not equal to] 0. Utilizing properties of these distributions, we propose a general family of distributions of Z such that HZ has the MACG ([Sigma]) distribution.

Keywords: Stiefel; manifolds; orientation; of; a; random; matrix; matrix; angular; Gaussian; distributions; matrix-variate; normal; distributions; matrix; elliptically; symmetric; distributions (search for similar items in EconPapers)
Date: 1990
References: Add references at CitEc
Citations: View citations in EconPapers (21)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/0047-259X(90)90050-R
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:jmvana:v:33:y:1990:i:2:p:265-274

Ordering information: This journal article can be ordered from
http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

Access Statistics for this article

Journal of Multivariate Analysis is currently edited by de Leeuw, J.

More articles in Journal of Multivariate Analysis from Elsevier
Bibliographic data for series maintained by Catherine Liu ().