This paper introduces a novel unified framework for testing mutual independence among a vector of random objects that may reside in different metric spaces, including some existing methodologies as special cases. The backbone of the proposed tests is the notion of joint distance profiles, which uniquely characterize the joint law of random objects under a mild condition on the joint law or on the metric spaces. Our test statistics measure 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, where flexible data-adaptive weight profiles are incorporated for power enhancement. We derive the limiting distribution of the test statistics under the null hypothesis of mutual independence and show that the proposed tests with specific weight profiles are asymptotically distribution-free if the marginal distance profiles are continuous. We also establish the consistency of the tests under sequences of alternative hypotheses converging to the null. Furthermore, since the asymptotic tests with non-trivial weight profiles require the knowledge of the underlying data distribution, we adopt a permutation scheme to approximate the $p$-values and provide theoretical guarantees that the permutation-based tests control the type I error rate under the null and are consistent under the alternatives. We demonstrate the power of the proposed tests across various types of data objects through simulations and real data applications, where our tests are shown to have superior performance compared with popular existing approaches.

翻译：本文提出了一种新颖的统一框架，用于检验可能位于不同度量空间中的随机对象向量之间的互独立性，该框架将若干现有方法作为特例包含在内。所提出检验的核心是联合距离剖面的概念，该概念在联合律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。我们的检验统计量通过引入灵活的数据自适应权重剖面以提升检验功效，衡量了每个数据点相对于随机对象向量的联合律与边缘律乘积的联合距离剖面差异。我们推导了原假设（互独立性）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检验是渐近分布自由的。我们还建立了检验在收敛于原假设的备择假设序列下的一致性。此外，由于具有非平凡权重剖面的渐近检验需要已知底层数据分布，我们采用置换方案来近似计算 $p$ 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率，在备择假设下具有一致性。我们通过模拟实验和实际数据应用，展示了所提出检验在处理各类数据对象时的效力，结果表明我们的检验相较于现有主流方法具有更优越的性能。