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arXiv · 1911.01985

Behavior of Fr\'echet mean and Central Limit Theorems on spheres

Do Tran
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Abstract

We compute higher derivatives of the Fr\'{e}chet function on spheres with an absolutely continuous and rotationally symmetric probability distribution. Consequences include (i)~a practical condition to test if the mode of the symmetric distribution is a local Fr\'{e}chet mean; (ii)~a Central Limit Theorem on spheres with practical assumptions and an explicit limiting distribution; and (iii)~an answer to the question of whether the smeary effect can occur on spheres with absolutely continuous and rotationally symmetric distributions: with the method presented here, it can in dimension at least~$4$.

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Do Tran. 2019-11-05. Behavior of Fr\'echet mean and Central Limit Theorems on spheres. https://arxiv.org/abs/1911.01985

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