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

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  • Hendriks, Harrie
  • Landsman, Zinoviy

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.

Suggested Citation

  • Hendriks, Harrie & Landsman, Zinoviy, 1998. "Mean Location and Sample Mean Location on Manifolds: Asymptotics, Tests, Confidence Regions," Journal of Multivariate Analysis, Elsevier, vol. 67(2), pages 227-243, November.
    • Handle: RePEc:eee:jmvana:v:67:y:1998:i:2:p:227-243
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    1. 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.
    2. Chikuse, Yasuko, 1990. "The matrix angular central Gaussian distribution," Journal of Multivariate Analysis, Elsevier, vol. 33(2), pages 265-274, May.
    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, 1996. "Asymptotic behavior of sample mean location for manifolds," Statistics & Probability Letters, Elsevier, vol. 26(2), pages 169-178, February.
    5. D. J. Best & N. I. Fisher, 1979. "Efficient Simulation of the von Mises Distribution," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 28(2), pages 152-157, June.
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    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. Rabi Bhattacharya & Rachel Oliver, 2019. "Nonparametric Analysis of Non-Euclidean Data on Shapes and Images," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 81(1), pages 1-36, February.
    4. 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.
    5. Ruite Guo & Hwiyoung Lee & Vic Patrangenaru, 2023. "Test for Homogeneity of Random Objects on Manifolds with Applications to Biological Shape Analysis," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(2), pages 1178-1204, August.
    6. Vic Patrangenaru & Mingfei Qiu & Marius Buibas, 2014. "Two Sample Tests for Mean 3D Projective Shapes from Digital Camera Images," Methodology and Computing in Applied Probability, Springer, vol. 16(2), pages 485-506, June.
    7. Benoit Ahanda & Daniel E. Osborne & Leif Ellingson, 2022. "Robustness of lognormal confidence regions for means of symmetric positive definite matrices when applied to mixtures of lognormal distributions," METRON, Springer;Sapienza Università di Roma, vol. 80(3), pages 281-303, December.
    8. 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;The Institute of Statistical Mathematics, vol. 64(6), pages 1227-1259, December.
    9. Wang, Yunfan & Patrangenaru, Vic & Guo, Ruite, 2020. "A Central Limit Theorem for extrinsic antimeans and estimation of Veronese–Whitney means and antimeans on planar Kendall shape spaces," Journal of Multivariate Analysis, Elsevier, vol. 178(C).
    10. Vic Patrangenaru & Yifang Deng, 2021. "Extrinsic Regression and Anti-Regression on Projective Shape Manifolds," Methodology and Computing in Applied Probability, Springer, vol. 23(2), pages 629-646, June.
    11. H. Fotouhi & M. Golalizadeh, 2015. "Highly resistant gradient descent algorithm for computing intrinsic mean shape on similarity shape spaces," Statistical Papers, Springer, vol. 56(2), pages 391-410, May.

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