标题： 高维均匀性检验在重尾备选假设下的渐近分析 显示英文标题

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

摘要： 我们研究了高维均匀性检验问题，即在给定 $n$ 个数据点位于 $p$-维单位超球面上的条件下，检验潜在分布是否为均匀分布。 虽然在固定 $p$ 的情况下该问题已被广泛研究，但在高维情形下仅存在三种检验程序：Rayleigh 检验 \cite{Cutting-P-V}、Bingham 检验 \cite{Cutting-P-V2} 和 packing 检验 \cite{Jiang13}。 大多数现有研究集中在这前两种检验上，packing 检验的一致性仍然是开放问题。 我们证明，在涉及重尾分布投影的一些替代类假设下，Rayleigh 检验渐近无能，Bingham 检验的渐近功效等价于随机猜测。 相比之下，我们从理论上表明 packing 检验对这些替代类假设具有强大功效，而经验研究表明其显著水平因极值统计量的收敛速度缓慢而出现严重扭曲。 通过利用这三种检验的渐近独立性，我们随后基于 Fisher 组合技术提出了一种新检验，结合了它们的优势。 新检验被证明具备每个个体检验的所有最优性质，并且与 packing 检验不同，它能够保持良好的第一类错误控制能力。 显示英文摘要

摘要： 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.