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

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.