We compute higher derivatives of the Fréchet 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 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.
A great debt goes to Stephan Huckemann and Benjamin Eltzner for enlightening conversations on smeariness on spheres. The author gratefully thanks his advisor, Ezra Miller, for invaluable advice and comments throughout the project. Thanks also go to the anonymous referees, and an associate editor for helpful suggestions to improve the paper.
The author was supported by grant Hu 1575/7-1 and for conference travel by grant NSF DMS-1702395.
"Behavior of the Fréchet mean and Central Limit Theorems on spheres." Braz. J. Probab. Stat. 35 (3) 590 - 608, August 2021. https://doi.org/10.1214/21-BJPS499