Finite Sample Smeariness on Spheres

TL;DR

This paper investigates Finite Sample Smeariness (FSS) on spheres, revealing its prevalence in various distributions and proposing bootstrap methods to correct its effects on statistical testing.

Contribution

It extends the understanding of FSS from circles to higher-dimensional spheres, showing all rotationally symmetric distributions exhibit FSS of Type I.

Findings

FSS occurs in all rotationally symmetric distributions on spheres.

Bootstrap tests can correct for FSS effects.

FSS of Type II may not occur on higher-dimensional spheres.

Abstract

Finite Sample Smeariness (FSS) has been recently discovered. It means that the distribution of sample Fr\'echet means of underlying rather unsuspicious random variables can behave as if it were smeary for quite large regimes of finite sample sizes. In effect classical quantile-based statistical testing procedures do not preserve nominal size, they reject too often under the null hypothesis. Suitably designed bootstrap tests, however, amend for FSS. On the circle it has been known that arbitrarily sized FSS is possible, and that all distributions with a nonvanishing density feature FSS. These results are extended to spheres of arbitrary dimension. In particular all rotationally symmetric distributions, not necessarily supported on the entire sphere feature FSS of Type I. While on the circle there is also FSS of Type II it is conjectured that this is not possible on higher-dimensional spheres.