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Intrinsic means on the circle: uniqueness, locus and asymptotics

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  • T. Hotz
  • S. Huckemann

Abstract

This paper gives a comprehensive treatment of local uniqueness, asymptotics and numerics for intrinsic sample means on the circle. It turns out that local uniqueness as well as rates of convergence are governed by the distribution near the antipode. If the distribution is locally less than uniform there, we have local uniqueness and asymptotic normality with a square-root rate. With increased proximity to the uniform distribution the rate can be arbitrarily slow, and in the limit, local uniqueness is lost. Further, we give general distributional conditions, e.g., unimodality, that ensure global uniqueness. Along the way, we discover that sample means can occur only at the vertices of a regular polygon which allows to compute intrinsic sample means in linear time from sorted data. This algorithm is finally applied in a simulation study demonstrating the dependence of the convergence rates on the behavior of the density at the antipode. Copyright The Institute of Statistical Mathematics, Tokyo 2015

Suggested Citation

  • T. Hotz & S. Huckemann, 2015. "Intrinsic means on the circle: uniqueness, locus and asymptotics," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(1), pages 177-193, February.
    • Handle: RePEc:spr:aistmt:v:67:y:2015:i:1:p:177-193
    DOI: 10.1007/s10463-013-0444-7
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    References listed on IDEAS

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    1. Kaziska, David & Srivastava, Anuj, 2008. "The Karcher mean of a class of symmetric distributions on the circle," Statistics & Probability Letters, Elsevier, vol. 78(11), pages 1314-1316, August.
    2. Stephan Huckemann, 2011. "Inference on 3D Procrustes Means: Tree Bole Growth, Rank Deficient Diffusion Tensors and Perturbation Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 38(3), pages 424-446, September.
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    Cited by:

    1. Stephan F. Huckemann, 2021. "Comments on: Recent advances in directional statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 71-75, March.
    2. Yannis Pantazis & Michail Tsagris & Andrew T. A. Wood, 2019. "Gaussian Asymptotic Limits for the α-transformation in the Analysis of Compositional Data," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 81(1), pages 63-82, February.
    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. Arthur Pewsey & Eduardo García-Portugués, 2021. "Recent advances in directional statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 1-58, March.

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