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

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

Received: 1 August 2011
Revised: 9 February 2012

Abstract

Let () be the unit circle in ℝ² 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 (). 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.