On the tight constant in the multivariate Dvoretzky–Kiefer–Wolfowitz inequality

Abstract

We derive the tight constant in the multivariate version of the Dvoretzky–Kiefer–Wolfowitz inequality. The inequality is leveraged to construct the first fully non-parametric test for multivariate probability distributions including a simple formula for the test statistic. We also generalize the test under appropriate α-mixing conditions and describe applications of the tests to machine learning and representative sampling.