On the power of axial tests of uniformity on spheres

TL;DR

This paper develops optimal statistical tests for uniformity on spheres under axial symmetry, addressing both known and unknown axes, and evaluates their performance through theoretical analysis and simulations.

Contribution

It introduces Le Cam optimal tests based on the sample covariance matrix for axial symmetry with known axis and studies the non-null behavior of classical tests for unknown axes.

Findings

Optimal tests are based on the sample covariance matrix.

Classical tests like Bingham have predictable non-null behavior.

Monte Carlo simulations confirm theoretical asymptotic results.

Abstract

Testing uniformity on the $p$-dimensional unit sphere is arguably the most fundamental problem in directional statistics. In this paper, we consider this problem in the framework of axial data, that is, under the assumption that the $n$ observations at hand are randomly drawn from a distribution that charges antipodal regions equally. More precisely, we focus on axial, rotationally symmetric, alternatives and first address the problem under which the direction $\theta$ of the corresponding symmetry axis is specified. In this setup, we obtain Le Cam optimal tests of uniformity, that are based on the sample covariance matrix (unlike their non-axial analogs, that are based on the sample average). For the more important unspecified-$\theta$ problem, some classical tests are available in the literature, but virtually nothing is known on their non-null behavior. We therefore study the non-null behavior of the celebrated Bingham test and of other tests that exploit the single-spiked nature of the considered alternatives. We perform Monte Carlo exercises to investigate the finite-sample behavior of our tests and to show their agreement with our asymptotic results.