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.
Similar content being viewed by others
References
Afsari, B. (2011). Riemannian \({L}^p\) center of mass: existence, uniqueness, and convexity. Proceedings of the American Mathematical Society, 139, 655–773.
Bhattacharya, R.N., Patrangenaru, V. (2003). Large sample theory of intrinsic and extrinsic sample means on manifolds I. The Annals of Statistics, 31(1), 1–29.
Bhattacharya, R.N., Patrangenaru, V. (2005). Large sample theory of intrinsic and extrinsic sample means on manifolds II. The Annals of Statistics, 33(3), 1225–1259.
Fisher, N. (1993). Statistical analysis of circular data. Cambridge: Cambridge University Press.
Fréchet, M. (1948). Les éléments aléatoires de nature quelconque dans un espace distancié. Annales de l’institut Henri Poincaré, 10(4), 215–310.
Huckemann, S. (2011). Inference on 3D Procrustes means: tree boles growth, rank-deficient diffusion tensors and perturbation models. Scandinavian Journal of Statistics, 38(3), 424–446.
Karcher, H. (1977). Riemannian center of mass and mollifier smoothing. Communications on Pure and Applied Mathematics, XXX, 509–541.
Kaziska, D., Srivastava, A. (2008). The Karcher mean of a class of symmetric distributions on the circle. Statistics and Probability Letters, 78(11), 1314–1316.
Kobayashi, S., Nomizu, K. (1969). Foundations of differential geometry (vol. II). Chichester: Wiley.
Le, H. (1998). On the consistency of Procrustean mean shapes. Advances of Applied Probability (SGSA), 30(1), 53–63.
Mardia, K., Patrangenaru, V. (2005). Directions and projective shapes. The Annals of Statistics, 33, 1666–1699.
Mardia, K. V., Jupp, P. E. (2000). Directional statistics. New York: Wiley.
McKilliam, R.G., Quinn, B.G., Clarkson, I.V.L. (2012). Direction estimation by minimum squared arc length. IEEE Transactions on Signal Processing, 60(5), 2115–2124.
Pennec, X. (2006). Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements. Journal of Mathematical Imaging and Vision, 25(1), 127–154.
Ziezold, H. (1977). Expected figures and a strong law of large numbers for random elements in quasi-metric spaces. Transaction of the 7th Prague Conference on Information Theory, Statistical Decision Function and Random Processes, A, 591–602.
Acknowledgments
The authors thank the two anonymous reviewers as well as the editorial committee for their comments which helped in improving this manuscript. T. Hotz acknowledges support by DFG CRC 803, S. Huckemann by DFG HU 1575/2-1 and the Niedersachsen Vorab of the Volkswagen Foundation. Both authors are more than grateful for support by SAMSI whose 2010 workshop on AOOD and the successive research stays there considerably enabled this research.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Hotz, T., Huckemann, S. Intrinsic means on the circle: uniqueness, locus and asymptotics. Ann Inst Stat Math 67, 177–193 (2015). https://doi.org/10.1007/s10463-013-0444-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-013-0444-7