On a projection-based class of uniformity tests on the hypersphere

TL;DR

This paper introduces a new class of projection-based uniformity tests on the hypersphere, extending classical circular tests and including the first Anderson-Darling-like test for hyperspherical data, with proven asymptotic properties and practical applications.

Contribution

It develops a novel, unified framework for uniformity testing on the hypersphere, generalizing existing tests and providing new tools with theoretical and empirical validation.

Findings

New tests extend classical circular tests to higher dimensions

The proposed tests are asymptotically distribution-free and locally optimal

Simulation studies show competitive performance against existing methods

Abstract

We propose a projection-based class of uniformity tests on the hypersphere using statistics that integrate, along all possible directions, the weighted quadratic discrepancy between the empirical cumulative distribution function of the projected data and the projected uniform distribution. Simple expressions for several test statistics are obtained for the circle and sphere, and relatively tractable forms for higher dimensions. Despite its different origin, the proposed class is shown to be related with the well-studied Sobolev class of uniformity tests. Our new class proves itself advantageous by allowing to derive new tests for hyperspherical data that neatly extend the circular tests by Watson, Ajne, and Rothman, and by introducing the first instance of an Anderson-Darling-like test for such data. The asymptotic distributions and the local optimality against certain alternatives of the new tests are obtained. A simulation study evaluates the theoretical findings and evidences that, for certain scenarios, the new tests are competitive against previous proposals. The new tests are employed in three astronomical applications.