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 nonparametric tests are based on the statistic T_n, which is the L₁ distance between the two empirical distributions restricted to a finite partition. Both tests reject the 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 distribution of the test statistic is obtained, leading to an asymptotically α-level test procedure.