Behavior of Fréchet mean and Central Limit Theorems on spheres

11/05/2019

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by   Do Tran, et al.

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Jacobi fields are used to compute higher derivatives of the Fréchet function on spheres with absolutely continuous rotationally symmetric probability distribution. Consequences include (i) a practical condition to test if the mode of the symmetric distribution is a local Fréchet 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.