The Annals of Statistics

Large sample theory of intrinsic and extrinsic sample means on manifolds

Rabi Bhattacharya and Vic Patrangenaru

Full-text: Open access


Sufficient conditions are given for the uniqueness of intrinsic and extrinsic means as measures of location of probability measures Q on Riemannian manifolds. It is shown that, when uniquely defined, these are estimated consistently by the corresponding indices of the empirical $\hat Q_n$. Asymptotic distributions of extrinsic sample means are derived. Explicit computations of these indices of $\hat Q_n$ and their asymptotic dispersions are carried out for distributions on the sphere $S^d$ (directional spaces), real projective space $\mathbb{R}P^{N-1}$ (axial spaces) and $\mathbb{C} P^{k-2}$ (planar shape spaces).

Article information

Ann. Statist., Volume 31, Number 1 (2003), 1-29.

First available in Project Euclid: 26 February 2003

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H11: Directional data; spatial statistics
Secondary: 62H10: Distribution of statistics

Fréchet mean intrinsic mean extrinsic mean consistency equivariant embedding mean planar shape


Bhattacharya, Rabi; Patrangenaru, Vic. Large sample theory of intrinsic and extrinsic sample means on manifolds. Ann. Statist. 31 (2003), no. 1, 1--29. doi:10.1214/aos/1046294456.

Export citation


  • AMARI, S.-I. (1985). Differential-Geometrical Methods in Statistics. Lecture Notes in Statist. 28. Springer, New York.
  • BARNDORFF-NIELSEN, O. E. and COX, D. R. (1994). Inference and Asy mptotics. Chapman and Hall, London.
  • BERAN, R. J. (1979). Exponential models for directional data. Ann. Statist. 7 1162-1178.
  • BERAN, R. and FISHER, N. I. (1998). Nonparametric comparison of mean directions or mean axes. Ann. Statist. 26 472-493.
  • BOOKSTEIN, F. L. (1991). Morphometric Tools for Landmark Data: Geometry and Biology. Cambridge Univ. Press.
  • BOOKSTEIN, F. L. and MARDIA, K. V. (2000). A family of EM-ty pe algorithms for missing morphometric data. In Abstracts of the 19th L.A.S.R Workshop (J. T. Kent and R. G. Ay kroy d, eds.).
  • BURBEA, J. and RAO, C. R. (1982). Entropy differential metric, distance and divergence measures in probability spaces: A unified approach. J. Multivariate Anal. 12 575-596.
  • CHAVEL, I. (1993). Riemannian Geometry: A Modern Introduction. Cambridge Univ. Press.
  • DO CARMO, M. P. (1992). Riemannian Geometry. Birkhäuser, Boston.
  • DRy DEN, I. L. and MARDIA, K. V. (1993). Multivariate shape analysis. Sankhy¯a Ser. A 55 460-480.
  • DRy DEN, I. L. and MARDIA, K. V. (1998). Statistical Shape Analy sis. Wiley, New York.
  • EFRON, B. (1975). Defining the curvature of a statistical problem (with applications to second order efficiency) (with discussion). Ann. Statist. 3 1189-1242.
  • EFRON, B. (1982). The Jackknife, the Bootstrap and Other Resampling Plans. SIAM, Philadelphia.
  • ÉMERY, M. and MOKOBODZKI, G. (1991). Sur le bary centre d'une probabilité dans une variété. Séminaire de Probabilités XXV. Lecture Notes in Math. 1485 220-233. Springer, Berlin.
  • FISHER, N. I. (1993). Statistical Analy sis of Circular Data. Cambridge Univ. Press.
  • FISHER, N. I. and HALL, P. (1992). Bootstrap methods for directional data. In The Art of Statistical Science: A Tribute to G. S. Watson (K. V. Mardia, ed.) 47-63. Wiley, New York.
  • FISHER, N. I., HALL, P., JING, B.-Y. and WOOD, A. T. A. (1996). Improved pivotal methods for constructing confidence regions with directional data. J. Amer. Statist. Assoc. 91 1062- 1070.
  • HELGASON, S. (1978). Differential Geometry, Lie Groups, and Sy mmetric Spaces. Academic Press, New York.
  • KARCHER, H. (1977). Riemannian center of mass and mollifier smoothing. Comm. Pure Appl. Math. 30 509-541.
  • KENDALL, D. G. (1984). Shape manifolds, Procrustean metrics, and complex projective spaces. Bull. London Math. Soc. 16 81-121.
  • KENDALL, D. G., BARDEN, D., CARNE, T. K. and LE, H. (1999). Shape and Shape Theory. Wiley, New York.
  • KENDALL, W. S. (1990). Probability, convexity, and harmonic maps with small image. I. Uniqueness and fine existence. Proc. London Math. Soc. 61 371-406.
  • KENT, J. T. (1992). New directions in shape analysis. In The Art of Statistical Science: A Tribute to G. S. Watson (K. V. Mardia, ed.) 115-128. Wiley, New York.
  • KENT, J. T. (1994). The complex Bingham distribution and shape analysis. J. Roy. Statist. Soc. Ser. B 56 285-299.
  • KENT, J. T. and MARDIA, K. V. (1997). Consistency of Procrustes estimators. J. Roy. Statist. Soc. Ser. B 59 281-290.
  • KOBAy ASHI, S. (1968). Isometric imbeddings of compact sy mmetric spaces. Tôhoku Math. J. 20 21-25.
  • KOBAy ASHI, S. (1972). Transformation Groups in Differential Geometry. Springer, New York.
  • KOBAy ASHI, S. and NOMIZU, K. (1996). Foundations of Differential Geometry 2. (Reprint of the 1969 original.) Wiley, New York.
  • LE, H. (1998). On the consistency of Procrustean mean shapes. Adv. in Appl. Probab. 30 53-63.
  • LE, H. and KUME, A. (2000). The Fréchet mean shape and the shape of means. Adv. in Appl. Probab. 32 101-113.
  • MARDIA, K. V. and JUPP, P. E. (1999). Directional Statistics. Wiley, New York.
  • MILNOR, J. (1963). Morse Theory. Princeton Univ. Press.
  • OLLER, J. M. and CORCUERA, J. M. (1995). Intrinsic analysis of statistical estimation. Ann. Statist. 23 1562-1581.
  • PRENTICE, M. J. (1984). A distribution-free method of interval estimation for unsigned directional data. Biometrika 71 147-154.
  • PRENTICE, M. J. and MARDIA, K. V. (1995). Shape changes in the plane for landmark data. Ann. Statist. 23 1960-1974.
  • RAO, C. R. (1945). Information and the accuracy attainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc. 37 81-91.
  • SIBSON, R. (1978). Studies in the robustness of multidimensional scaling: Procrustes analysis. J. Roy. Statist. Soc. Ser. B 40 234-238.
  • WANG, H.-C. (1952). Two-point homogeneous spaces. Ann. Math. 55 177-191.
  • WARNER, F. W. (1965). The conjugate locus of a Riemannian manifold. Amer. J. Math. 87 575-604.
  • WATSON, G. S. (1983). Statistics on Spheres. Wiley, New York.
  • ZIEZOLD, H. (1977). 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 A 591-602. Reidel, Dordrecht.