One and two sample Dvoretzky-Kiefer-Wolfowitz-Massart type inequalities for differing underlying distributions

TL;DR

This paper extends the Dvoretzky-Kiefer-Wolfowitz-Massart inequality to handle cases where the underlying distributions differ, providing bounds for the KS-distance even when the null hypothesis of equality does not hold.

Contribution

It introduces a new inequality applicable to scenarios with different underlying distributions, broadening the scope of KS-based statistical tests.

Findings

Derived inequalities for differing distributions

Applicable to one and two sample cases

Enhances the theoretical foundation of KS tests

Abstract

Kolmogorov-Smirnov (KS) tests rely on the convergence to zero of the KS-distance $d(F_n,G)$ in the one sample case, and of $d(F_n,G_m)$ in the two sample case. In each case the assumption (the null hypothesis) is that $F=G$, and so $d(F,G)=0$. In this paper we extend the Dvoretzky-Kiefer-Wolfowitz-Massart inequality to also apply to cases where $F \neq G$, i.e. when it is possible that $d(F,G) > 0$.