Paired Sample Tests for High-dimensional Uncorrelatedness via Random Integration

2026-06-14

AI Summary

This study addresses the challenge of nonparametric testing for independence between high-dimensional random vectors, particularly in settings where both dimensionality and sample size grow simultaneously. The authors propose a novel test statistic based on a weighted squared L2 norm of the sample covariance matrix, constructed via an extension of random integrals. This approach yields an asymptotically standard normal distribution without requiring assumptions about the relative growth rates of dimension and sample size. The method demonstrates enhanced power in detecting "weak but pervasive" dependence structures while maintaining accurate type I error control. Extensive Monte Carlo simulations confirm its favorable finite-sample performance, and its practical utility is illustrated through an application to the analysis of associations between DNA methylation and gene expression data.

Abstract

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.

Problem

• high-dimensional uncorrelatedness
• paired sample test
• nonparametric test
• covariance matrix
• random integration

Innovation

• high-dimensional uncorrelatedness
• random integration
• nonparametric test
• asymptotic normality
• L2 norm of covariance