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Concentrated matrix Langevin distributions

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  • Chikuse, Yasuko

Abstract

This paper concerns the matrix Langevin distributions, exponential-type distributions defined on the two manifolds of our interest, the Stiefel manifold Vk,m and the manifold Pk,m-k of mxm orthogonal projection matrices idempotent of rank k which is equivalent to the Grassmann manifold Gk,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.

Suggested Citation

  • Chikuse, Yasuko, 2003. "Concentrated matrix Langevin distributions," Journal of Multivariate Analysis, Elsevier, vol. 85(2), pages 375-394, May.
  • Handle: RePEc:eee:jmvana:v:85:y:2003:i:2:p:375-394
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    References listed on IDEAS

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    1. Chikuse, Yasuko, 1998. "Density Estimation on the Stiefel Manifold," Journal of Multivariate Analysis, Elsevier, vol. 66(2), pages 188-206, August.
    2. Chikuse, Y., 1993. "High Dimensional Asymptotic Expansions for the Matrix Langevin Distributions on the Stiefel Manifold," Journal of Multivariate Analysis, Elsevier, vol. 44(1), pages 82-101, January.
    3. Mardia, K. V. & Khatri, C. G., 1977. "Uniform distribution on a Stiefel manifold," Journal of Multivariate Analysis, Elsevier, vol. 7(3), pages 468-473, September.
    4. Chikuse, Yasuko, 1990. "Distributions of orientations on Stiefel manifolds," Journal of Multivariate Analysis, Elsevier, vol. 33(2), pages 247-264, May.
    5. Chikuse, Y. & Watson, G. S., 1995. "Large Sample Asymptotic Theory of Tests for Uniformity on the Grassmann Manifold," Journal of Multivariate Analysis, Elsevier, vol. 54(1), pages 18-31, July.
    6. Constantine, A. G. & Muirhead, R. J., 1976. "Asymptotic expansions for distributions of latent roots in multivariate analysis," Journal of Multivariate Analysis, Elsevier, vol. 6(3), pages 369-391, September.
    7. Watson, Geoffrey S., 1984. "The theory of concentrated Langevin distributions," Journal of Multivariate Analysis, Elsevier, vol. 14(1), pages 74-82, February.
    8. Chikuse, Yasuko, 1991. "Asymptotic expansions for distributions of the large sample matrix resultant and related statistics on the Stiefel manifold," Journal of Multivariate Analysis, Elsevier, vol. 39(2), pages 270-283, November.
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    Cited by:

    1. Davy Paindaveine & Thomas Verdebout, 2019. "Inference for Spherical Location under High Concentration," Working Papers ECARES 2019-02, ULB -- Universite Libre de Bruxelles.

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