Abstract: In this paper, we propose a new homogeneous test for two highd-imensional random vectors. Our test is built on a new measure, the so-called characteristic distance, which can completely characterize the homogeneity of two distributions. The newly proposed metric has some desirable properties, for example, it possesses a clear and intuitive probabilistic interpretation, and can be used to address the high-dimensional distance inference. Theoretically, the limiting behaviors under the conventional fixed dimension and high-dimensional distance inference are thoroughly investigated. Simulation studies and real data analysis are presented to illustrate the finite-sample performance of the proposed test statistic.

Key words and phrases: Characteristic distance, high dimensionality, permutation procedure, test of homogeneity, U-statistic.