Many high-dimensional hypothesis tests aim to globally examine marginal or low-dimensional features of a high-dimensional joint distribution, such as testing of mean vectors, covariance matrices and regression coefficients. This paper constructs a family of U-statistics as unbiased estimators of the $\ell_{p}$-norms of those features. We show that under the null hypothesis, the U-statistics of different finite orders are asymptotically independent and normally distributed. Moreover, they are also asymptotically independent with the maximum-type test statistic, whose limiting distribution is an extreme value distribution. Based on the asymptotic independence property, we propose an adaptive testing procedure which combines $p$-values computed from the U-statistics of different orders. We further establish power analysis results and show that the proposed adaptive procedure maintains high power against various alternatives.

中文翻译：

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高维测试中渐近独立的 U 统计量


许多高维假设检验旨在全局检查高维联合分布的边际或低维特征，例如均值向量、协方差矩阵和回归系数的检验。本文构建了一系列 U 统计量作为这些特征的 $\ell_{p}$-范数的无偏估计量。我们证明，在原假设下，不同有限阶的 U 统计量是渐近独立且呈正态分布的。此外，它们还与最大型检验统计量渐近独立，其极限分布是极值分布。基于渐进独立性，我们提出了一种自适应测试程序，该程序结合了从不同阶的 U 统计量计算出的 $p$ 值。我们进一步建立功效分析结果，并表明所提出的自适应程序相对于各种替代方案保持高功效。