Statistics on Riemannian manifolds: asymptotic distribution and curvature

Authors:
Abhishek Bhattacharya and Rabi Bhattacharya

Journal:
Proc. Amer. Math. Soc. **136** (2008), 2959-2967

MSC (2000):
Primary 62G20; Secondary 62E20, 62H35

Published electronically:
March 14, 2008

MathSciNet review:
2399064

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**1.**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.**2.**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**, 10.1214/aos/1046294456**3.**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**, 10.1214/009053605000000093**4.**Fred L. Bookstein,*Morphometric tools for landmark data*, Cambridge University Press, Cambridge, 1997. Geometry and biology; Reprint of the 1991 original. MR**1469220****5.**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****6.**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****7.**N. I. Fisher, T. Lewis, and B. J. J. Embleton,*Statistical analysis of spherical data*, Cambridge University Press, Cambridge, 1987. MR**899958****8.**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**, 10.1006/jmva.1998.1776**9.**H. Karcher,*Riemannian center of mass and mollifier smoothing*, Comm. Pure Appl. Math.**30**(1977), no. 5, 509–541. MR**0442975****10.**Jürgen Jost,*Riemannian geometry and geometric analysis*, 4th ed., Universitext, Springer-Verlag, Berlin, 2005. MR**2165400****11.**David G. Kendall,*Shape manifolds, Procrustean metrics, and complex projective spaces*, Bull. London Math. Soc.**16**(1984), no. 2, 81–121. MR**737237**, 10.1112/blms/16.2.81**12.**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**, 10.1112/plms/s3-61.2.371**13.**Huiling Le,*Locating Fréchet means with application to shape spaces*, Adv. in Appl. Probab.**33**(2001), no. 2, 324–338. MR**1842295**, 10.1239/aap/999188316**14.**John M. Lee,*Riemannian manifolds*, Graduate Texts in Mathematics, vol. 176, Springer-Verlag, New York, 1997. An introduction to curvature. MR**1468735****15.**Kanti V. Mardia and Vic Patrangenaru,*Directions and projective shapes*, Ann. Statist.**33**(2005), no. 4, 1666–1699. MR**2166559**, 10.1214/009053605000000273**16.**Xavier Pennec,*Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements*, J. Math. Imaging Vision**25**(2006), no. 1, 127–154. MR**2254442**, 10.1007/s10851-006-6228-4

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
62G20,
62E20,
62H35

Retrieve articles in all journals with MSC (2000): 62G20, 62E20, 62H35

Additional Information

**Abhishek Bhattacharya**

Affiliation:
Department of Mathematics, University of Arizona, Tucson, Arizona 85721

Email:
abhishek@math.arizona.edu

**Rabi Bhattacharya**

Affiliation:
Department of Mathematics, University of Arizona, Tucson, Arizona 85721

Email:
rabi@math.arizona.edu

DOI:
https://doi.org/10.1090/S0002-9939-08-09445-8

Keywords:
Intrinsic mean,
shape space of k-ads,
nonparametric analysis

Received by editor(s):
July 15, 2007

Published electronically:
March 14, 2008

Additional Notes:
This research was supported by NSF Grant DMS 04-06143

Communicated by:
Edward C. Waymire

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.