Abstract
This paper is concerned with the study of a circular random distribution called geodesic normal distribution recently proposed for general manifolds. This distribution, parameterized by two real numbers associated to some specific location and dispersion concepts, looks like a standard Gaussian on the real line except that the support of this variable is [0, 2π) and that the Euclidean distance is replaced by the geodesic distance on the circle. Some properties are studied and comparisons with the von Mises distribution in terms of intrinsic and extrinsic means and variances are provided. Finally, the problem of estimating the parameters through the maximum likelihood method is investigated and illustrated with some simulations.
Similar content being viewed by others
References
Bhattacharya R, Patrangenaru V (2003) Large sample theory of intrinsic and extrinsic sample means on manifolds: I. Ann Stat 31(1): 1–29
Bhattacharya R, Patrangenaru V (2005) Large sample theory of intrinsic and extrinsic sample means on manifolds: II. Ann Stat 33: 1225–1259
Do Carmo MP (1976) Differential geometry of curves and surfaces, vol 2. Prentice-Hall, Englewood Cliffs
Huckemann S, Hotz T, Munk A (2010) Intrinsic shape analysis: geodesic PCA for Riemannian manifolds modulo isometric lie group actions. Stat Sinica 20(1): 1–58
Jammalamadaka SR, Sengupta AA (2001) Topics in circular statistics. World Scientific Pub Co Inc, Singapore
Karcher H (1977) Riemannian center of mass and mollifier smoothing. Commun Pure Appl Math 30(5): 509–541
Kaziska D, Srivastava A (2008) The Karcher mean of a class of symmetric distributions on the circle. Stat Probab Lett 78(11): 1314–1316
Lehmann EL, Casella G (1998) Theory of point estimation. Springer, Berlin
Le H (1998) On the consistency of Procrustean mean shapes. Adv Appl Probab 30(1): 53–63
Le H (2001) Locating Fréchet means with application to shape spaces. Adv Appl Probab 33(2): 324–338
Mardia KV (1972) Statistics of directional data. Academic Press, London
Mardia KV, Jupp PE (2000) Directional statistics. Wiley, Chichester
Pennec X (2006) Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements. J Math Imaging Vis 25(1): 127–154
Van der Vaart AW (1998) Asymptotic statistics. Cambridge University Press, Cambridge
Watson GS (1983) Statistics on spheres. Wiley, London
Ziezold H (1977) On expected figures and a strong law of large numbers for random elements in quasi- metric spaces. In: Transactions on 7th Prague conference information theory, statistics decision function, random processes A, pp 591–602
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Coeurjolly, JF., Bihan, N.L. Geodesic normal distribution on the circle. Metrika 75, 977–995 (2012). https://doi.org/10.1007/s00184-011-0363-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-011-0363-7