Technological advances have enabled us to collect a lot of complex data objects, where homogeneity structure among these objects is widely used in Statistics. However, the existing metrics of homogeneity are subject to some qualifications, such as assumptions about the moment and parameters. To overcome the limitation, this paper first introduces the characteristic distance, a novel metric that entirely characterizes the homogeneity of two distributions. The proposed distance possesses some desirable statistical properties: (i) It is a distribution-free or, more commonly, nonparametric test, thus is robust to the data; (ii) It is nonnegative and equal to zero if and only if the two distributions are homogeneous; (iii) The novel measure possesses a clear and intuitive probabilistic interpretation, moreover, its empirical version is easy to calculate and can be reduced to a sum of two V-statistics. Theoretically, the asymptotic distributions, including the mixture of \(\chi ^{2}\) distributions under the null hypothesis and the asymptotic normality of the alternative hypothesis are thoroughly investigated. Simulation studies and a real data application suggest that the empirical characteristic distance has a preferable power in detecting the homogeneity of distributions.

技术进步使我们能够收集大量复杂的数据对象，这些对象之间的同质结构在统计学中被广泛使用。然而，现有的同质性指标受到一些限制，例如关于时刻和参数的假设。为了克服这个限制，本文首先介绍了特征距离，这是一种完全表征两个分布同质性的新度量。所提出的距离具有一些理想的统计特性：（i）它是无分布的或更常见的非参数检验，因此对数据具有鲁棒性；(ii) 当且仅当两个分布是齐次的，它是非负的并且等于零；(iii) 新测度具有清晰直观的概率解释，此外，它的经验版本很容易计算，可以简化为两个 V 统计量的总和。理论上，渐近分布，包括对原假设下的\(\chi ^{2}\)分布和备择假设的渐近正态性进行了深入研究。模拟研究和实际数据应用表明，经验特征距离在检测分布的同质性方面具有更好的能力。