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

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 2736346 | Zbl 1220.53040

[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 1962498 | Zbl 1020.62026

[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 1705458 | Zbl 0948.53020

[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 113758 | Numdam | MR 1187782 | Zbl 0753.60046

[6] N.I. Fisher, Statistical analysis of circular data. Cambridge University Press, Cambridge (1993). | MR 1251957 | Zbl 0788.62047

[7] M. Fréchet, Les éléments aléatoires de nature quelconque dans un espace distancié. Ann. Inst. Henri Poincaré 10 (1948) 215-310. | EuDML 79021 | Numdam | MR 27464 | Zbl 0035.20802

[8] H. Karcher, Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30 (1977) 509-541. | MR 442975 | Zbl 0354.57005

[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 2444322 | Zbl 1144.62323

[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 1891212 | Zbl 0940.60006

[11] H. Le, On the consistency of procrustean mean shapes. Adv. Appl. Probab. 30 (1998) 53-63. | MR 1618880 | Zbl 0906.60007

[12] H. Le, Locating Fréchet means with application to shape spaces. Adv. Appl. Probab. 33 (2001) 324-338. | MR 1842295 | Zbl 0990.60008

[13] H. Le, Estimation of Riemannian barycentres. LMS J. Comput. Math. 7 (2004) 193-200. | MR 2085875 | Zbl 1054.60011

[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 1828667 | Zbl 0935.62065

[15] P. Massart, The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality. Ann. Probab. 18 (1990) 1269-1283. | MR 1062069 | Zbl 0713.62021

[16] J.M. Oller and J.M. Corcuera, Intrinsic analysis of statistical estimation. Ann. Statist. 23 (1995) 1562-1581. | MR 1370296 | Zbl 0843.62027

[17] X. Pennec, Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements. J. Math. Imaging Vision 25 (2006) 127-154. | MR 2254442

[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 501230 | Zbl 0413.60024