The Annals of Statistics

A smeary central limit theorem for manifolds with application to high-dimensional spheres

Benjamin Eltzner and Stephan F. Huckemann

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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.

Article information

Ann. Statist., Volume 47, Number 6 (2019), 3360-3381.

Received: January 2018
Revised: August 2018
First available in Project Euclid: 31 October 2019

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Mathematical Reviews number (MathSciNet)

Primary: 62G20: Asymptotic properties 62H11: Directional data; spatial statistics
Secondary: 53C22: Geodesics [See also 58E10] 58K45: Singularities of vector fields, topological aspects

Fréchet means asymptotic consistency and normality asymptotics on manifolds lower asymptotic rate high dimension low sample size


Eltzner, Benjamin; Huckemann, Stephan F. A smeary central limit theorem for manifolds with application to high-dimensional spheres. Ann. Statist. 47 (2019), no. 6, 3360--3381. doi:10.1214/18-AOS1781.

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