Dvoretzky--Kiefer--Wolfowitz Inequalities for the Two-sample Case

TL;DR

This paper investigates the validity of the Dvoretzky--Kiefer--Wolfowitz inequality in the two-sample case, establishing conditions under which the inequality holds with the optimal constant and identifying exceptions through theoretical analysis and computational experiments.

Contribution

It extends the DKWM inequality to the two-sample case, determining the exact sample size thresholds for the inequality to hold with the sharp constant and identifying specific exceptions.

Findings

The DKWM inequality holds for n ≥ 458 when m=n.

For n<458, the inequality holds with a constant greater than 2.

The inequality generally holds for n ≥ 4 and m<n up to 200, with specific cases verified.

Abstract

The Dvoretzky--Kiefer--Wolfowitz (DKW) inequality says that if $F_n$ is an empirical distribution function for variables i.i.d.\ with a distribution function $F$, and $K_n$ is the Kolmogorov statistic $\sqrt{n}\sup_x|(F_n-F)(x)|$, then there is a finite constant $C$ such that for any $M>0$, $\Pr(K_n>M) \leq C\exp(-2M^2).$ Massart proved that one can take C=2 (DKWM inequality) which is sharp for $F$ continuous. We consider the analogous Kolmogorov--Smirnov statistic $KS_{m,n}$ for the two-sample case and show that for $m=n$, the DKW inequality holds with C=2 if and only if $n\geq 458$. For $n_0\leq n<458$ it holds for some $C>2$ depending on $n_0$. For $m\neq n$, the DKWM inequality fails for the three pairs $(m,n)$ with $1\leq m < n\leq 3$. We found by computer search that for $n\geq 4$, the DKWM inequality always holds for $1\leq m< n\leq 200$, and further that it holds for $n=2m$ with $101\leq m\leq 300$. We conjecture that the DKWM inequality holds for pairs $m\leq n$ with the $457+3 =460$ exceptions mentioned.