Testing for mutual independence among several random vectors is a challenging problem, and in recent years, it has gained significant attention in statistics and machine learning literature. Most of the existing tests of independence deal with only two random vectors, and they do not have straightforward generalizations for testing mutual independence among more than two random vectors of arbitrary dimensions. On the other hand, there are various tests for mutual independence among several random variables, but these univariate tests do not have natural multivariate extensions. In this article, we propose two general recipes, one based on inter-point distances and the other based on linear projections, for multivariate extensions of these univariate tests. Under appropriate regularity conditions, these resulting tests turn out to be consistent whenever we have consistency for the corresponding univariate tests. We carry out extensive numerical studies to compare the empirical performance of these proposed methods with the state-of-the-art methods.

测试几个随机向量之间的相互独立性是一个具有挑战性的问题，近年来，它已在统计学和机器学习文献中引起了广泛关注。现有的大多数独立性测试仅涉及两个随机向量，并且它们没有直接的概括来测试两个以上任意维度的随机向量之间的相互独立性。另一方面，对于多个随机变量之间的相互独立性，有各种检验，但是这些单变量检验没有自然的多元扩展。在本文中，我们提出了两种通用方法，一种基于点间距离，另一种基于线性投影，用于这些单变量检验的多变量扩展。在适当的规律性条件下，只要我们对相应的单变量检验具有一致性，这些结果检验就证明是一致的。我们进行了广泛的数值研究，以比较这些提议的方法与最新方法的经验性能。