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April 2013 Minimax properties of Fréchet means of discretely sampled curves
Jérémie Bigot, Xavier Gendre
Ann. Statist. 41(2): 923-956 (April 2013). DOI: 10.1214/13-AOS1104

Abstract

We study the problem of estimating a mean pattern from a set of similar curves in the setting where the variability in the data is due to random geometric deformations and additive noise. We propose an estimator based on the notion of Fréchet mean that is a generalization of the standard notion of averaging to non-Euclidean spaces. We derive a minimax rate for this estimation problem, and we show that our estimator achieves this optimal rate under the asymptotics where both the number of curves and the number of sampling points go to infinity.

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Jérémie Bigot. Xavier Gendre. "Minimax properties of Fréchet means of discretely sampled curves." Ann. Statist. 41 (2) 923 - 956, April 2013. https://doi.org/10.1214/13-AOS1104

Information

Published: April 2013
First available in Project Euclid: 29 May 2013

zbMATH: 1360.62169
MathSciNet: MR3099126
Digital Object Identifier: 10.1214/13-AOS1104

Subjects:
Primary: 62G08
Secondary: 62G20

Keywords: Curve registration , deformable models , Fréchet mean , Functional data analysis , lie group action , minimax rate of convergence , non-Euclidean metric , Sobolev balls

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 2 • April 2013
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