Abstract We treat the problem of testing mutual independence of k high-dimensional random vectors when the data are multivariate normal and k ≥ 2 is a fixed integer. For this purpose, we focus on the vector correlation coefficient, ρ V and propose an extension of its classical estimator which is constructed to correct potential sources of inconsistency related to the high dimensionality. Building on the proposed estimator of ρ V , we derive the new test statistic and study its limiting behavior in a general high-dimensional asymptotic framework which allows the vector’s dimensionality arbitrarily exceed the sample size. Specifically, we show that the asymptotic distribution of the test statistic under the main hypothesis of independence is standard normal and that the proposed test is size and power consistent. Using our statistics, we further construct the step-down multiple comparison procedure based on the closed testing strategy for the simultaneous test for independence. Accuracy of the proposed tests in finite samples is shown through simulations for a variety of high-dimensional scenarios in combination with a number of alternative dependence structures. Real data analysis is performed to illustrate the utility of the test procedures.

中文翻译：

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测试高维变量的独立性：基于 ρV 系数的方法

摘要 我们研究了当数据为多元正态且k≥2为固定整数时k个高维随机向量的相互独立性检验问题。为此，我们关注向量相关系数 ρ V 并提出其经典估计器的扩展，该估计器旨在纠正与高维相关的潜在不一致源。基于提出的 ρ V 估计量，我们推导出新的检验统计量并研究其在通用高维渐近框架中的极限行为，该框架允许向量的维数任意超过样本大小。具体来说，我们表明在独立性主要假设下检验统计量的渐近分布是标准正态的，并且建议的检验是大小和功效一致的。使用我们的统计数据，我们进一步构建了基于封闭测试策略的降压多重比较程序，用于同时测试独立性。有限样本中提议的测试的准确性通过对各种高维场景的模拟与许多替代依赖结构相结合来显示。执行实际数据分析以说明测试程序的效用。