This paper develops a novel unified framework for testing mutual independence among random objects residing in possibly different metric spaces. The framework generalizes existing methodologies and introduces new measures of mutual independence, and proposes associated tests that achieve minimax rate optimality and exhibit strong empirical power. The foundation of the proposed tests is the new concept of joint distance profiles, which uniquely characterize the joint law of random objects under a mild condition on either the joint law or the metric spaces. Our test statistics quantify the difference of the joint distance profiles of each data point with respect to the joint law and the product of marginal laws of the vector of random objects. To enhance power, we consider integrating this difference with respect to different measures and incorporate flexible data-adaptive weight profiles in the test statistics. We derive the limiting distribution of the test statistics under the null hypothesis of mutual independence and show that the proposed tests with certain weight profiles are asymptotically distribution-free if the marginal distance profiles are continuous. Furthermore, we establish the consistency of the tests under sequences of alternative hypotheses converging to the null. For practical implementations, we employ a permutation scheme to approximate the $p$-values and provide theoretical guarantees that the permutation-based tests maintain type I error control under the null and achieve consistency under alternatives. We demonstrate the power of the proposed tests across various types of data objects through simulations and real data applications, where the new tests exhibit better performance compared with popular existing approaches.

翻译：本文提出了一种新颖的统一框架，用于检验可能存在于不同度量空间中的随机对象之间的互独立性。该框架推广了现有方法，引入了新的互独立性度量，并提出了达到极小极大率最优性且展现出强大经验功效的关联检验方法。所提出检验的基础是联合距离剖面这一新概念，该概念在联合分布或度量空间满足温和条件时，能够唯一刻画随机对象的联合分布律。我们的检验统计量量化了每个数据点相对于随机对象向量的联合分布律与边缘分布律乘积的联合距离剖面之间的差异。为提升检验功效，我们考虑将该差异关于不同测度进行积分，并在检验统计量中引入灵活的数据自适应权重剖面。我们推导了原假设（互独立）下检验统计量的极限分布，并证明当边缘距离剖面连续时，采用特定权重剖面的所提出检验是渐近分布自由的。此外，我们在收敛于原假设的一系列备择假设下建立了检验的一致性。对于实际应用，我们采用置换方案来近似 $p$ 值，并从理论上保证了基于置换的检验在原假设下能控制第一类错误，并在备择假设下达到一致性。我们通过模拟和实际数据应用，展示了所提出检验在处理各类数据对象时的强大功效，其中新检验方法相较于现有流行方法表现出更优的性能。