Necessary and sufficient condition for the existence of a Fréchet mean on the circle

Charlier, Benjamin

ESAIM: Probability and Statistics, Tome 17 (2013), pp. 635-649.

Let ( ${𝕊}^1,d_{{𝕊}^1}$ S 1 , d S 1 ) be the unit circle in ℝ² endowed with the arclength distance. We give a sufficient and necessary condition for a general probability measure μ to admit a well defined Fréchet mean on ( ${𝕊}^1,d_{{𝕊}^1}$ S 1 , d S 1 ). We derive a new sufficient condition of existence P(α, ϕ) with no restriction on the support of the measure. Then, we study the convergence of the empirical Fréchet mean to the Fréchet mean and we give an algorithm to compute it.

[1] B. Afsari, Riemannian Lp center of mass: existence, uniqueness, and convexity. Proc. Amer. Math. Soc. 139 (2011) 655-673. | MR | Zbl

[2] R. Bhattacharya and V. Patrangenaru, Large sample theory of intrinsic and extrinsic sample means on manifolds, I. Ann. Stat. 31 (2003) 1-29. | MR | Zbl

[3] R.S. Buss and J.P. Fillmore, Spherical averages and applications to spherical splines and interpolation. ACM Trans. Graph. 20 (2001) 95-126.

[4] J.M. Corcuera and W.S. Kendall, Riemannian barycentres and geodesic convexity. Math. Proc. Cambridge Philos. Soc. 127 (1999) 253-269. | MR | Zbl

[5] M. Émery and G. Mokobodzki, Sur le barycentre d'une probabilité dans une variété, in Séminaire de Probabilités, XXV, vol. 1485 of Lect. Notes Math. (1991) 220-233. | EuDML | Numdam | MR | Zbl

[6] N.I. Fisher, Statistical analysis of circular data. Cambridge University Press, Cambridge (1993). | MR | Zbl

[7] M. Fréchet, Les éléments aléatoires de nature quelconque dans un espace distancié. Ann. Inst. Henri Poincaré 10 (1948) 215-310. | EuDML | Numdam | MR | Zbl

[8] H. Karcher, Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30 (1977) 509-541. | MR | Zbl

[9] D. Kaziska and A. Srivastava, The karcher mean of a class of symmetric distributions on the circle. Stat. Probab. Lett. 78 (2008) 1314-1316 (2008). | MR | Zbl

[10] D.G. Kendall, D. Barden, T.K. Carne and H. Le, Shape and shape theory. Wiley Series in Probability and Statistics. John Wiley & Sons Ltd., Chichester (1999). | MR | Zbl

[11] H. Le, On the consistency of procrustean mean shapes. Adv. Appl. Probab. 30 (1998) 53-63. | MR | Zbl

[12] H. Le, Locating Fréchet means with application to shape spaces. Adv. Appl. Probab. 33 (2001) 324-338. | MR | Zbl

[13] H. Le, Estimation of Riemannian barycentres. LMS J. Comput. Math. 7 (2004) 193-200. | MR | Zbl

[14] K.V. Mardia and P.E. Jupp, Directional statistics. Wiley Series in Probability and Statistics. John Wiley & Sons Ltd., Chichester (2000). Revised reprint of ıt Statistics of directional data by Mardia [ MR0336854 (49 #1627)]. | MR | Zbl

[15] P. Massart, The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality. Ann. Probab. 18 (1990) 1269-1283. | MR | Zbl

[16] J.M. Oller and J.M. Corcuera, Intrinsic analysis of statistical estimation. Ann. Statist. 23 (1995) 1562-1581. | MR | Zbl

[17] X. Pennec, Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements. J. Math. Imaging Vision 25 (2006) 127-154. | MR

[18] H. Ziezold, On expected figures and a strong law of large numbers for random elements in quasi-metric spaces, in Transactions of the Seventh Prague Conference on Information Theory, Statistical Decision Functions, Random Processes and of the Eighth European Meeting of Statisticians (Tech. Univ. Prague, Prague, 1974). Reidel, Dordrecht (1977) 591-602. | MR | Zbl