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

Abstract

Let $({\S }^1,d_{{\S }^1})$ be the unit circle in ${\mathbb{R}}^2$ endowed with the arclength distance. We give a sufficient and necessary condition for a general probability measure $\mu$ to admit a well defined Fréchet mean on $({\S }^1,d_{{\S }^1})$. %This criterion allows to recover already known sufficient conditions of existence. We derive a new sufficient condition of existence $P(\alpha ,\phi )$ 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.

arXiv commentFirst submission : Advances in Applied Probability (AAP) on May 17th 2011 (ref. AP/13983)