The Annals of Statistics

Large sample theory of intrinsic and extrinsic sample means on manifolds—II

Rabi Bhattacharya and Vic Patrangenaru

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Abstract

This article develops nonparametric inference procedures for estimation and testing problems for means on manifolds. A central limit theorem for Fréchet sample means is derived leading to an asymptotic distribution theory of intrinsic sample means on Riemannian manifolds. Central limit theorems are also obtained for extrinsic sample means w.r.t. an arbitrary embedding of a differentiable manifold in a Euclidean space. Bootstrap methods particularly suitable for these problems are presented. Applications are given to distributions on the sphere Sd (directional spaces), real projective space ℝPN−1 (axial spaces), complex projective space ℂPk−2 (planar shape spaces) w.r.t. Veronese–Whitney embeddings and a three-dimensional shape space Σ34.

Article information

Source
Ann. Statist. Volume 33, Number 3 (2005), 1225-1259.

Dates
First available in Project Euclid: 1 July 2005

Permanent link to this document
https://projecteuclid.org/euclid.aos/1120224101

Digital Object Identifier
doi:10.1214/009053605000000093

Mathematical Reviews number (MathSciNet)
MR2195634

Zentralblatt MATH identifier
1072.62033

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

Keywords
Fréchet mean extrinsic mean central limit theorem confidence regions bootstrapping

Citation

Bhattacharya, Rabi; Patrangenaru, Vic. Large sample theory of intrinsic and extrinsic sample means on manifolds—II. Ann. Statist. 33 (2005), no. 3, 1225--1259. doi:10.1214/009053605000000093. https://projecteuclid.org/euclid.aos/1120224101


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