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Density Estimation on the Stiefel Manifold

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  • Chikuse, Yasuko

Abstract

This paper develops the theory of density estimation on the Stiefel manifoldVk, m, whereVk, mis represented by the set ofm-kmatricesXsuch thatX'X=Ik, thek-kidentity matrix. The density estimation by the method of kernels is considered, proposing two classes of kernel density estimators with small smoothing parameter matrices and for kernel functions of matrix argument. Asymptotic behavior of various statistical measures of the kernel density estimators is investigated for small smoothing parameter matrix and/or for large sample size. Some decompositions of the Stiefel manifoldVk, mplay useful roles in the investigation, and the general discussion is applied and examined for a special kernel function. Alternative methods of density estimation are suggested, using decompositions ofVk, m.

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  • Chikuse, Yasuko, 1998. "Density Estimation on the Stiefel Manifold," Journal of Multivariate Analysis, Elsevier, vol. 66(2), pages 188-206, August.
    • Handle: RePEc:eee:jmvana:v:66:y:1998:i:2:p:188-206
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    References listed on IDEAS

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    1. 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.
    2. Chikuse, Yasuko, 1990. "The matrix angular central Gaussian distribution," Journal of Multivariate Analysis, Elsevier, vol. 33(2), pages 265-274, May.
    3. Chikuse, Yasuko, 1990. "Distributions of orientations on Stiefel manifolds," Journal of Multivariate Analysis, Elsevier, vol. 33(2), pages 247-264, May.
    4. 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.
    5. 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.
    6. 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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    2. Chikuse, Yasuko, 2003. "Concentrated matrix Langevin distributions," Journal of Multivariate Analysis, Elsevier, vol. 85(2), pages 375-394, May.

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