Behavior of the Fréchet mean and central limit theorems on spheres

DOI10.1214/21-BJPS499zbMATH Open1474.60060arXiv1911.01985OpenAlexW3188609997WikidataQ113751974 ScholiaQ113751974MaRDI QIDQ2244848

Do Tran

Publication date: 12 November 2021

Published in: Brazilian Journal of Probability and Statistics (Search for Journal in Brave)

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~$4$.

Full work available at URL: https://arxiv.org/abs/1911.01985

zbMATH Keywords

central limit theoremFréchet meansmearyspherical statistic

Mathematics Subject Classification ID

Directional data; spatial statistics (62H11) Asymptotic properties of nonparametric inference (62G20) Central limit and other weak theorems (60F05)

Cites Work

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• Statistics on Riemannian manifolds: asymptotic distribution and curvature
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• On the measure of the cut locus of a Fréchet mean

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