Testing independence with high-dimensional correlated samples
针对高维相关样本的独立性检验问题，提出一种计算简单且无需调参的检验统计量，并证明其极小极大最优性；同时开发了相关样本下Pearson相关系数的“三明治估计量”，用于控制多重检验中的错误发现率。
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.