Testing uniformity on high-dimensional spheres against monotone rotationally symmetric alternatives

TL;DR

This paper investigates the problem of testing uniformity on high-dimensional spheres against rotationally symmetric alternatives, deriving asymptotic optimality results and extending classical tests to high dimensions.

Contribution

It establishes two LAN structures for rotationally symmetric alternatives, proving the local optimality of the Rayleigh test in high dimensions and extending its power analysis.

Findings

High-dimensional Rayleigh test is asymptotically most powerful invariant.

Derived asymptotic non-null distribution of the Rayleigh test.

Results strengthen Rayleigh test's optimality in low dimensions.

Abstract

We consider the problem of testing uniformity on high-dimensional unit spheres. We are primarily interested in non-null issues. We show that rotationally symmetric alternatives lead to two Local Asymptotic Normality (LAN) structures. The first one is for fixed modal location $\theta$ and allows to derive locally asymptotically most powerful tests under specified $\theta$. The second one, that addresses the Fisher-von Mises-Langevin (FvML) case, relates to the unspecified-$\theta$ problem and shows that the high-dimensional Rayleigh test is locally asymptotically most powerful invariant. Under mild assumptions, we derive the asymptotic non-null distribution of this test, which allows to extend away from the FvML case the asymptotic powers obtained there from Le Cam's third lemma. Throughout, we allow the dimension $p$ to go to infinity in an arbitrary way as a function of the sample size $n$. Some of our results also strengthen the local optimality properties of the Rayleigh test in low dimensions. We perform a Monte Carlo study to illustrate our asymptotic results. Finally, we treat an application related to testing for sphericity in high dimensions.