On the asymptotic properties of a nonparametric L1-test statistic of homogeneity

We present two simple and explicit procedures for testing homogeneity of two independent multivariate samples of size n. The non-parametric tests are based on the statistic T-n, which is the L-1 distance between the two empirical distributions restricted to a finite partition. Both tests reject the null hypothesis of homogeneity if T-n becomes large, i.e., if T-n exceeds a threshold. We first discuss Chernoff-type large deviation properties of T-n. This results in a distribution-free strong consistent test of homogeneity. Then the asymptotic null distribution of the test statistic is obtained, leading to an asymptotically alpha-level test procedure.