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Mean Location and Sample Mean Location on Manifolds: Asymptotics, Tests, Confidence Regions

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  • Hendriks, Harrie
  • Landsman, Zinoviy
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    Abstract

    In a previous investigation we studied some asymptotic properties of the sample mean location on submanifolds of Euclidean space. The sample mean location generalizes least squares statistics to smooth compact submanifolds of Euclidean space. In this paper these properties are put into use. Tests for hypotheses about mean location are constructed and confidence regions for mean location are indicated. We study the asymptotic distribution of the test statistic. The problem of comparing mean locations for two samples is analyzed. Special attention is paid to observations on Stiefel manifolds including the orthogonal groupO(p) and spheresSk-1, and special orthogonal groupsSO(p). The results also are illustrated with our experience with simulations.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 67 (1998)
    Issue (Month): 2 (November)
    Pages: 227-243

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    Handle: RePEc:eee:jmvana:v:67:y:1998:i:2:p:227-243

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    Keywords: Mean location degenerate normal distribution sample Weingarten mapping cut-locus Stiefel manifold test confidence interval asymptotic acceptance-rejection method simulation;

    References

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    1. Chikuse, Yasuko, 1990. "The matrix angular central Gaussian distribution," Journal of Multivariate Analysis, Elsevier, vol. 33(2), pages 265-274, May.
    2. Hendriks, Harrie & Landsman, Zinoviy, 1996. "Asymptotic behavior of sample mean location for manifolds," Statistics & Probability Letters, Elsevier, vol. 26(2), pages 169-178, February.
    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. Hendriks, Harrie & Landsman, Zinoviy & Ruymgaart, Frits, 1996. "Asymptotic Behavior of Sample Mean Direction for Spheres," Journal of Multivariate Analysis, Elsevier, vol. 59(2), pages 141-152, November.
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    Cited by:
    1. Stephan Huckemann, 2012. "On the meaning of mean shape: manifold stability, locus and the two sample test," Annals of the Institute of Statistical Mathematics, Springer, vol. 64(6), pages 1227-1259, December.
    2. Huckemann, Stephan & Hotz, Thomas, 2009. "Principal component geodesics for planar shape spaces," Journal of Multivariate Analysis, Elsevier, vol. 100(4), pages 699-714, April.
    3. Osborne, Daniel & Patrangenaru, Vic & Ellingson, Leif & Groisser, David & Schwartzman, Armin, 2013. "Nonparametric two-sample tests on homogeneous Riemannian manifolds, Cholesky decompositions and Diffusion Tensor Image analysis," Journal of Multivariate Analysis, Elsevier, vol. 119(C), pages 163-175.

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