Measuring and testing homogeneity of distributions by characteristic distance

Abstract

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.