This paper proposes a novel nonparametric test to assess the uncorrelatedness between two high-dimensional random vectors. We develop our test by generalizing the random integration proposed by Jiang et al. (2023, 2024), and the resulting test statistic estimates a weighted squared $\mathscr{L}_2$ norm of the covariance matrix. Asymptotic properties of the test statistic are derived by letting both the sample size $n$ and the dimension $p$ diverge to infinity. Under the null hypothesis of uncorrelatedness, our proposed test statistic is asymptotically normal with zero mean and unit variance, without requiring any specification of the relative magnitude regarding $n$ and $p$. Monte Carlo simulations demonstrate the good finite-sample performance of our proposed methods. Compared with many existing tests, our test statistic is more powerful at detecting ``weak but pervasive'' dependence while maintaining a comparable empirical size. The advantages of the proposed methods are further illustrated by an empirical analysis that assesses the correlation between DNA methylation and gene expression.

翻译：本文提出了一种新颖的非参数检验方法，用于评估两个高维随机向量之间的不相关性。我们通过推广Jiang等人（2023, 2024）提出的随机积分方法来构建检验统计量，该统计量估计了协方差矩阵的加权平方$\mathscr{L}_2$范数。通过让样本量$n$和维度$p$同时趋向无穷大，推导了检验统计量的渐近性质。在原假设（不相关性）下，所提出的检验统计量渐近服从均值为零、方差为1的正态分布，且无需指定$n$和$p$的相对大小。蒙特卡洛模拟表明所提方法具有良好的有限样本性能。与现有多种检验相比，我们的检验统计量在检测“弱而普遍”的依赖关系时具有更强的统计功效，同时保持可比较的经验尺寸。通过一项评估DNA甲基化与基因表达之间相关性的实证分析，进一步展示了所提方法的优势。