Statistics on Riemannian manifolds: asymptotic distribution and curvature

Bhattacharya, Abhishek; Bhattacharya, Rabi

doi:10.1090/S0002-9939-08-09445-8

Statistics on Riemannian manifolds: asymptotic distribution and curvature
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by Abhishek Bhattacharya and Rabi Bhattacharya PDF

Proc. Amer. Math. Soc. 136 (2008), 2959-2967 Request permission

Abstract:

In this article a nonsingular asymptotic distribution is derived for a broad class of underlying distributions on a Riemannian manifold in relation to its curvature. Also, the asymptotic dispersion is explicitly related to curvature. These results are applied and further strengthened for the planar shape space of k-ads.

References

A. Bhattacharya and R. Bhattacharya, Nonparametric Statistics on Manifolds with Applications to Shape Spaces. In Pushing the Limits of Contemporary Statistics: Contributions in Honor of J. K. Ghosh, IMS Lecture Series (S. Ghoshal and B. Clarke, eds.), 2008.
Rabi Bhattacharya and Vic Patrangenaru, Large sample theory of intrinsic and extrinsic sample means on manifolds. I, Ann. Statist. 31 (2003), no. 1, 1–29. MR 1962498, DOI 10.1214/aos/1046294456
Rabi Bhattacharya and Vic Patrangenaru, Large sample theory of intrinsic and extrinsic sample means on manifolds. II, Ann. Statist. 33 (2005), no. 3, 1225–1259. MR 2195634, DOI 10.1214/009053605000000093
Fred L. Bookstein, Morphometric tools for landmark data, Cambridge University Press, Cambridge, 1997. Geometry and biology; Reprint of the 1991 original. MR 1469220
Manfredo Perdigão do Carmo, Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992. Translated from the second Portuguese edition by Francis Flaherty. MR 1138207, DOI 10.1007/978-1-4757-2201-7
I. L. Dryden and K. V. Mardia, Statistical shape analysis, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 1998. MR 1646114
N. I. Fisher, T. Lewis, and B. J. J. Embleton, Statistical analysis of spherical data, Cambridge University Press, Cambridge, 1987. MR 899958, DOI 10.1017/CBO9780511623059
Harrie Hendriks and Zinoviy Landsman, Mean location and sample mean location on manifolds: asymptotics, tests, confidence regions, J. Multivariate Anal. 67 (1998), no. 2, 227–243. MR 1659156, DOI 10.1006/jmva.1998.1776
H. Karcher, Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math. 30 (1977), no. 5, 509–541. MR 442975, DOI 10.1002/cpa.3160300502
Jürgen Jost, Riemannian geometry and geometric analysis, 4th ed., Universitext, Springer-Verlag, Berlin, 2005. MR 2165400
David G. Kendall, Shape manifolds, Procrustean metrics, and complex projective spaces, Bull. London Math. Soc. 16 (1984), no. 2, 81–121. MR 737237, DOI 10.1112/blms/16.2.81
Wilfrid S. Kendall, Probability, convexity, and harmonic maps with small image. I. Uniqueness and fine existence, Proc. London Math. Soc. (3) 61 (1990), no. 2, 371–406. MR 1063050, DOI 10.1112/plms/s3-61.2.371
Huiling Le, Locating Fréchet means with application to shape spaces, Adv. in Appl. Probab. 33 (2001), no. 2, 324–338. MR 1842295, DOI 10.1239/aap/999188316
John M. Lee, Riemannian manifolds, Graduate Texts in Mathematics, vol. 176, Springer-Verlag, New York, 1997. An introduction to curvature. MR 1468735, DOI 10.1007/b98852
Kanti V. Mardia and Vic Patrangenaru, Directions and projective shapes, Ann. Statist. 33 (2005), no. 4, 1666–1699. MR 2166559, DOI 10.1214/009053605000000273
Xavier Pennec, Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements, J. Math. Imaging Vision 25 (2006), no. 1, 127–154. MR 2254442, DOI 10.1007/s10851-006-6228-4