On a $L_1$-Test Statistic of Homogeneity

Abstract

We present a simple and explicit multivariate procedure for testing homogeneity of two independent samples of size $n$. The test statistic $T_n$ is the $L_1$ distance between the two empirical distributions restricted to a finite partition. We first discuss Chernoff-type large deviation properties of $T_n$. This results in a distribution-free strongly consistent test of homogeneity, which rejects the null if $T_n$ becomes large. Then the asymptotic null distribution of the test statistic is obtained, leading to a new consistent test procedure.