This paper is concerned with testing global null hypotheses about population mean vectors of high-dimensional data. Current tests require either strong mixing (independence) conditions on the individual components of the high-dimensional data or high-order moment conditions. In this paper, we propose a novel class of bootstrap hypothesis tests based on $\ell_p$-statistics with $p \in [1, \infty]$ which requires neither of these assumptions. We study asymptotic size, unbiasedness, consistency, and Bahadur slope of these tests. Capitalizing on these theoretical insights, we develop a modified bootstrap test with improved power properties and a self-normalized bootstrap test for elliptically distributed data. We then propose two novel bias correction procedures to improve the accuracy of the bootstrap test in finite samples, which leverage measure concentration and hypercontractivity properties of $\ell_p$-norms in high dimensions. Numerical experiments support our theoretical results in finite samples.

翻译：本文研究高维数据总体均值向量的全局原假设检验问题。现存的检验方法要么要求高维数据各分量之间具有强混合（独立性）条件，要么需要高阶矩条件。本文提出一类基于$\ell_p$统计量（$p \in [1, \infty]$）的新型自助法假设检验，该检验无需上述两种假设。我们研究了这些检验的渐近水平、无偏性、相合性及Bahadur斜率。基于这些理论洞见，我们发展了具有改进功效特性的修正自助法检验，以及适用于椭圆分布数据的自标准化自助法检验。随后提出两种新颖的偏差校正程序，利用高维$\ell_p$范数的测度集中性与超收缩性质，提升有限样本下自助法检验的准确性。数值实验在有限样本情境下验证了我们的理论结果。