High dimensional asymptotic expansions for the matrix Langevin distributions on the Stiefel manifold
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.
matrix uniform and Langevin distributions
scientific article; zbMATH DE number 168195