A Test of Homogeneity for Two Multivariate Populations

The classical tests of homogeneity, such as the twosample Kolmogorov-Smirnov test, do not have a natural extension to comparing two multivariate populations. G. J. Székely and N. K. Bakirov have proposed a new test based on Euclidean distance between sample elements. This test can be applied to testing homogeneity of any two multivariate populations with finite second moments, and the test is rotation invariant and consistent. We discuss the theoretical background of the proposed test, and show that a practical implementation is possible via nonparametric bootstrap for the composite hypothesis of equal distributions when both distributions are unspecified. Empirical results in the univariate case suggest that this test compares favorably with existing tests. We also present empirical results for multivariate distributions.