Let ( S 1 , d S 1 ) 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 ( S 1 , d S 1 ). 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.
@article{PS_2013__17__635_0, author = {Charlier, Benjamin}, title = {Necessary and sufficient condition for the existence of a {Fr\'echet} mean on the circle}, journal = {ESAIM: Probability and Statistics}, pages = {635--649}, publisher = {EDP-Sciences}, volume = {17}, year = {2013}, doi = {10.1051/ps/2012015}, mrnumber = {3126155}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/ps/2012015/} }
TY - JOUR AU - Charlier, Benjamin TI - Necessary and sufficient condition for the existence of a Fréchet mean on the circle JO - ESAIM: Probability and Statistics PY - 2013 SP - 635 EP - 649 VL - 17 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps/2012015/ DO - 10.1051/ps/2012015 LA - en ID - PS_2013__17__635_0 ER -
%0 Journal Article %A Charlier, Benjamin %T Necessary and sufficient condition for the existence of a Fréchet mean on the circle %J ESAIM: Probability and Statistics %D 2013 %P 635-649 %V 17 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps/2012015/ %R 10.1051/ps/2012015 %G en %F PS_2013__17__635_0
Charlier, Benjamin. Necessary and sufficient condition for the existence of a Fréchet mean on the circle. ESAIM: Probability and Statistics, Volume 17 (2013), pp. 635-649. doi : 10.1051/ps/2012015. http://archive.numdam.org/articles/10.1051/ps/2012015/
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