Smeariness Begets Finite Sample Smeariness

TL;DR

This paper investigates finite sample smeariness in Fréchet means on curved spaces, revealing its prevalence and impact on statistical methods, especially on spheres, tori, and circles, and explores its relation to curvature.

Contribution

It demonstrates that finite sample smeariness occurs regularly on common curved spaces and establishes directional smeariness under curvature bounds, advancing understanding of Fréchet means in non-Euclidean geometry.

Findings

FSS occurs regularly on circles, tori, and spheres.

FSS affects the applicability of tangent space-based tests.

Near smeary distributions, FSS probability distributions always exist.

Abstract

Fr\'echet means are indispensable for nonparametric statistics on non-Euclidean spaces. For suitable random variables, in some sense, they "sense" topological and geometric structure. In particular, smeariness seems to indicate the presence of positive curvature. While smeariness may be considered more as an academical curiosity, occurring rarely, it has been recently demonstrated that finite sample smeariness (FSS) occurs regularly on circles, tori and spheres and affects a large class of typical probability distributions. FSS can be well described by the modulation measuring the quotient of rescaled expected sample mean variance and population variance. Under FSS it is larger than one - that is its value on Euclidean spaces - and this makes quantile based tests using tangent space approximations inapplicable. We show here that near smeary probability distributions there are always FSS probability distributions and as a first step towards the conjecture that all compact spaces feature smeary distributions, we establish directional smeariness under curvature bounds.