BALL DIVERGENCE: NONPARAMETRIC TWO SAMPLE TEST.In this paper, we first introduce Ball Divergence, a novel measure of the difference between two probability measures in separable Banach spaces, and show that the Ball Divergence of two probability measures is zero if and only if these two probability measures are identical without any moment assumption.  Using Ball Divergence, we present a metric rank test procedure to detect the equality of distribution measures underlying independent samples.  We show that this multivariate two sample test statistic is consistent with the Ball Divergence, and it converges to a mixture of χ2 distributions under the null hypothesis and a normal distribution under the alternative hypothesis. 
BALL DIVERGENCE
BALL DIVERGENCE: NONPARAMETRIC TWO SAMPLE TEST.In this paper, we first introduce Ball Divergence, a novel measure of the difference between two probability measures in separable Banach spaces, and show that the Ball Divergence of two probability measures is zero if and only if these two probability measures are identical without any moment assumption.  Using Ball Divergence, we present a metric rank test procedure to detect the equality of distribution measures underlying independent samples.  We show that this multivariate two sample test statistic is consistent with the Ball Divergence, and it converges to a mixture of χ2 distributions under the null hypothesis and a normal distribution under the alternative hypothesis. 
Ball Divergence
We show that this multivariate two sample test statistic is consistent with the Ball Divergence, and it converges to a mixture of χ2 distributions under the null hypothesis and a normal distribution under the alternative hypothesis. 
χ2 distributions
We show that this multivariate two sample test statistic is consistent with the Ball Divergence, and it converges to a mixture of χ2 distributions under the null hypothesis and a normal distribution under the alternative hypothesis. 
normal distribution