High Dimensional Asymptotic Expansions for the Matrix Langevin Distributions on the Stiefel Manifold

Y. Chikuse

Journal of Multivariate Analysis, 1993, vol. 44, issue 1, 82-101

Abstract: Let Vk,m denote the Stiefel manifold whose elements are m - k (m >= k) matrices X such that X'X = Ik. We may be interested in high dimensional (as m --> [infinity]) asymptotic behaviors of statistics on Vk,m. High dimensional Stiefel manifolds may appear in a geometrical study in other contexts, e.g., for the analysis of compositional data with an arbitrary number m of components. We consider the matrix Langevin L(m, k; F) and L(m, k; m1/2F) distributions, each with the singular value decomposition F = [Gamma] [Delta][Theta]' of an m - k parameter matrix F, where [Gamma] [set membership, variant] Vp,m, [Theta] [set membership, variant] Vp,k, and [Delta] = diag([lambda]1, ..., [lambda]p), [lambda]j > 0. For a random matrix X having each of the two distributions, we derive asymptotic expansions, for large m, for the probability density functions of the matrix variates Y = m1/2[Gamma]'X and W = YY' and of the related functions y = tr MY' /(tr MM')1/2 and w = tr W. Here M is an arbitrary p - k constant matrix. Putting [Delta] = 0 in the asymptotic expansions yields those for the uniform distribution. The asymptotic expansions derived in this paper may be useful for statistical inference on Vk,m.

Date: 1993
References: Add references at CitEc
Citations: View citations in EconPapers (2)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0047-259X(83)71005-5
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:44:y:1993:i:1:p:82-101

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 ().