# Testing independence with high-dimensional correlated samples

## TL;DR

This paper introduces a simple, tuning-free test for independence among high-dimensional correlated samples, proves its optimality, and applies it to improve large-scale multiple testing of correlations.

## Contribution

It proposes a novel independence test statistic for high-dimensional data with correlated samples and develops a de-correlation method for more accurate multiple testing.

## Key findings

- The test statistic has a known limiting null distribution.
- The method achieves minimax optimality in power.
- The de-correlation approach effectively controls false discovery rates.

## Abstract

Testing independence among a number of (ultra) high-dimensional random samples is a fundamental and challenging problem. By arranging $n$ identically distributed $p$-dimensional random vectors into a $p \times n$ data matrix, we investigate the problem of testing independence among columns under the matrix-variate normal modeling of data. We propose a computationally simple and tuning-free test statistic, characterize its limiting null distribution, analyze the statistical power and prove its minimax optimality. As an important by-product of the test statistic, a ratio-consistent estimator for the quadratic functional of a covariance matrix from correlated samples is developed. We further study the effect of correlation among samples to an important high-dimensional inference problem --- large-scale multiple testing of Pearson's correlation coefficients. Indeed, blindly using classical inference results based on the assumed independence of samples will lead to many false discoveries, which suggests the need for conducting independence testing before applying existing methods. To address the challenge arising from correlation among samples, we propose a "sandwich estimator" of Pearson's correlation coefficient by de-correlating the samples. Based on this approach, the resulting multiple testing procedure asymptotically controls the overall false discovery rate at the nominal level while maintaining good statistical power. Both simulated and real data experiments are carried out to demonstrate the advantages of the proposed methods.