Testing for independence between two random vectors is a fundamental problem in statistics. It is observed from empirical studies that many existing omnibus consistent tests may not work well for some strongly nonmonotonic and nonlinear relationships. To explore the reasons behind this issue, we novelly transform the multivariate independence testing problem equivalently into checking the equality of two bivariate means. An important observation we made is that the power loss is mainly due to cancellation of positive and negative terms in dependence metrics, making them very close to zero. Motivated by this observation, we propose a class of consistent metrics with a positive integer $\gamma$ that exactly characterize independence. Theoretically, we show that the metrics with even and infinity $\gamma$ can effectively avoid the cancellation, and have high powers under the alternatives that two mean differences offset each other. Since we target at a wide range of dependence scenarios in practice, we further suggest to combine the p-values of test statistics with different $\gamma$'s through the Fisher's method. We illustrate the advantages of our proposed tests through extensive numerical studies.

翻译：检验两个随机向量之间的独立性是统计学中的一个基本问题。实证研究表明，许多现有的全能一致性检验可能对某些强非单调和非线性关系失效。为探究这一问题的深层原因，我们创新性地将多变量独立性检验问题等价转化为检验两个双变量均值是否相等。一个重要发现是，功效损失主要源于依赖性度量中正负项的相互抵消，导致其值趋近于零。基于这一观察，我们提出了一类以正整数$\gamma$参数化的、能够精确刻画独立性的连续性度量。理论证明，偶数和无穷大$\gamma$对应的度量能有效避免抵消效应，并在两个均值差异相互抵消的备择假设下展现出高检验功效。鉴于实际中需要应对广泛的依赖性场景，我们进一步建议通过Fisher方法对不同$\gamma$值的检验统计量的p值进行整合。通过大量数值研究，我们验证了所提方法的优越性。