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

Benjamin Charlier (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

Let $(S^1,d_{S^1})$ be the unit circle in $\R^2$ 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 also derive a new sufficient condition without restriction on the support of the measure. Then, we study the convergence of the empirical Fréchet mean to the Fréchet mean. An algorithm to compute the empirical Fréchet mean is also given.

Mots clés: circular data; Fréchet mean; uniqueness

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Benjamin Charlier. Necessary and sufficient condition for the existence of a Fréchet mean on the circle. 2011. ⟨hal-00620965v1⟩