## The Annals of Statistics

### Large sample theory of intrinsic and extrinsic sample means on manifolds

#### Abstract

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

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

Dates
First available in Project Euclid: 26 February 2003

http://projecteuclid.org/euclid.aos/1046294456

Digital Object Identifier
doi:10.1214/aos/1046294456

Mathematical Reviews number (MathSciNet)
MR1962498

Zentralblatt MATH identifier
1020.62026

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

#### Citation

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. http://projecteuclid.org/euclid.aos/1046294456.

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• BLOOMINGTON, INDIANA 47405 DEPARTMENT OF MATHEMATICS AND STATISTICS GEORGIA STATE UNIVERSITY
• ATLANTA, GEORGIA 30303 E-MAIL: matvnp@geomstat.cs.gsu.edu