Asymptotic expansions for distributions of the large sample matrix resultant and related statistics on the Stiefel manifold

Abstract

Let V k, m denote the Stiefel manifold which consists of m × k( m ≥ k) matrices X such that X′ X = I k . Let X 1,…, X n be a random sample of size n from the matrix Langevin (or von Mises-Fisher) distribution on V k, m , which has the density proportional to exp(tr F′ X), with F an m × k matrix, and let Z = ( m n ) 1 2 Σ j = 1 n X j . The exact expression of the distribution of Z in an integral form is intractable. In this paper, we derive asymptotic expansions, for large n and up to the order of n −3, for the distributions of Z, Z′ Z, and related statistics in connection with testing problems on F, under the hypothesis of uniformity ( F = 0) and local alternative hypotheses. In the derivation, we utilize zonal and invariant polynomials in matrix arguments and Hermite and Laguerre polynomials in one-dimensional variable and matrix argument.