Asymptotic analysis of high-dimensional uniformity tests under heavy-tailed alternatives

TL;DR

This paper analyzes the limitations of existing high-dimensional uniformity tests under heavy-tailed alternatives and proposes a new combined test that leverages their strengths for improved power and error control.

Contribution

It provides a theoretical and empirical comparison of three tests, reveals their limitations, and introduces a novel Fisher combination test with optimal properties.

Findings

Rayleigh test is asymptotically blind under heavy-tailed alternatives

Bingham test has power equivalent to random guessing in these scenarios

The new combined test maintains good error control and high power

Abstract

We study the high-dimensional uniformity testing problem, which involves testing whether the underlying distribution is the uniform distribution, given $n$ data points on the $p$-dimensional unit hypersphere. While this problem has been extensively studied in scenarios with fixed $p$, only three testing procedures are known in high-dimensional settings: the Rayleigh test \cite{Cutting-P-V}, the Bingham test \cite{Cutting-P-V2}, and the packing test \cite{Jiang13}. Most existing research focuses on the former two tests, and the consistency of the packing test remains open. We show that under certain classes of alternatives involving projections of heavy-tailed distributions, the Rayleigh test is asymptotically blind, and the Bingham test has asymptotic power equivalent to random guessing. In contrast, we show theoretically that the packing test is powerful against such alternatives, and empirically that its size suffers from severe distortion due to the slow convergence nature of extreme-value statistics. By exploiting the asymptotic independence of these three tests, we then propose a new test based on Fisher's combination technique that combines their strengths. The new test is shown to enjoy all the optimality properties of each individual test, and unlike the packing test, it maintains excellent type-I error control.