A stereographic test of spherical uniformity

TL;DR

This paper presents a new stereographic-based statistical test for assessing uniformity on spheres, deriving its distribution and demonstrating superior performance in scenarios with antipodal dependence.

Contribution

Introduces a novel stereographic projection-based test for spherical uniformity, with explicit distribution derivation and improved power in specific dependence scenarios.

Findings

Test outperforms existing methods with antipodal dependence

Closed-form test statistic derived using Gegenbauer polynomials

Simulation results confirm asymptotic properties

Abstract

We introduce a test of uniformity for (hyper)spherical data motivated by the stereographic projection. The closed-form expression of the test statistic and its null asymptotic distribution are derived using Gegenbauer polynomials. The power against rotationally symmetric local alternatives is provided, and simulations illustrate the non-null asymptotic results. The stereographic test outperforms other tests in a testing scenario with antipodal dependence between observations.