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Necessary and sufficient condition for the existence of a Fréchet mean on the circle

Published online by Cambridge University Press:  04 November 2013

Benjamin Charlier
Benjamin Charlier*
Affiliation:
Institut de Mathématiques de Toulouse Université de Toulouse et CNRS (UMR 5219), F-31062 Toulouse, France. benjamin.charlier@math.univ-toulouse.fr
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Abstract

Let (\hbox{$\mathbb{S}^1, d_{\mathbb{S}^1}$}S1,dS1) be the unit circle inℝ2 endowed with the arclength distance. We give a sufficient and necessary condition for a general probability measure μ to admit a well defined Fréchet mean on (\hbox{$\mathbb{S}^1,d_{\mathbb{S}^1}$}S1,dS1). We derive a new sufficient condition of existence P(α, ϕ) with no restriction on the support of the measure. Then, we study the convergence of the empirical Fréchet mean to the Fréchet mean and we give an algorithm to compute it.

Keywords

Circular dataFréchet meanuniqueness
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 635 - 649
Copyright
© EDP Sciences, SMAI, 2013

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