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December 2019 A smeary central limit theorem for manifolds with application to high-dimensional spheres
Benjamin Eltzner, Stephan F. Huckemann
Ann. Statist. 47(6): 3360-3381 (December 2019). DOI: 10.1214/18-AOS1781

Abstract

The (CLT) central limit theorems for generalized Fréchet means (data descriptors assuming values in manifolds, such as intrinsic means, geodesics, etc.) on manifolds from the literature are only valid if a certain empirical process of Hessians of the Fréchet function converges suitably, as in the proof of the prototypical BP-CLT [Ann. Statist. 33 (2005) 1225–1259]. This is not valid in many realistic scenarios and we provide for a new very general CLT. In particular, this includes scenarios where, in a suitable chart, the sample mean fluctuates asymptotically at a scale $n^{\alpha }$ with exponents $\alpha <1/2$ with a nonnormal distribution. As the BP-CLT yields only fluctuations that are, rescaled with $n^{1/2}$, asymptotically normal, just as the classical CLT for random vectors, these lower rates, somewhat loosely called smeariness, had to date been observed only on the circle. We make the concept of smeariness on manifolds precise, give an example for two-smeariness on spheres of arbitrary dimension, and show that smeariness, although “almost never” occurring, may have serious statistical implications on a continuum of sample scenarios nearby. In fact, this effect increases with dimension, striking in particular in high dimension low sample size scenarios.

Citation

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Benjamin Eltzner. Stephan F. Huckemann. "A smeary central limit theorem for manifolds with application to high-dimensional spheres." Ann. Statist. 47 (6) 3360 - 3381, December 2019. https://doi.org/10.1214/18-AOS1781

Information

Received: 1 January 2018; Revised: 1 August 2018; Published: December 2019
First available in Project Euclid: 31 October 2019

Digital Object Identifier: 10.1214/18-AOS1781

Subjects:
Primary: 62G20, 62H11
Secondary: 53C22, 58K45

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 6 • December 2019
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