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Concentrated matrix Langevin distributions. (English) Zbl 1016.62065

Summary: This paper concerns the matrix Langevin distributions, exponential-type distributions defined on the two manifolds of our interest, the Stiefel manifold \(V_{k,m}\) and the manifold \(P_{k,m-k}\) of \(m\times m\) orthogonal projection matrices idempotent of rank \(k\) which is equivalent to the Grassmann manifold \(G_{k,m-k}\).
Asymptotic theorems are derived when the concentration parameters of the distributions are large. We investigate the asymptotic behavior of distributions of some (matrix) statistics constructed based on the sample mean matrices in connection with testing hypotheses of the orientation parameters, and obtain asymptotic results in the estimation of large concentration parameters and in the classification of the matrix Langevin distributions.

MSC:

62H10 Multivariate distribution of statistics
62E20 Asymptotic distribution theory in statistics
62H15 Hypothesis testing in multivariate analysis
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