DVORETZKY–KIEFER–WOLFOWITZ INEQUALITIES FOR THE TWO-SAMPLE CASE

Abstract

The Dvoretzky–Kiefer–Wolfowitz (DKW) inequality says that if Fn is an empirical distribution function for variables i.i.d. with a distribution function F, and Kn is the Kolmogorov statistic View the MathML source, then there is a constant C such that for any M>0, Pr(Kn>M)≤Cexp(−2M2). Massart proved that one can take C=2 (DKWM inequality), which is sharp for F continuous. We consider the analogous Kolmogorov–Smirnov statistic for the two-sample case and show that for m=n, the DKW inequality holds for n≥n0 for some C depending on n0, with C=2 if and only if n0≥458.

The DKWM inequality fails for the three pairs (m,n) with 1≤m<n≤3. We found by computer search that the inequality always holds for n≥4 if 1≤m<n≤200, and further for n=2m if 101≤m≤300. We conjecture that the DKWM inequality holds for all pairs m≤n with the 457+3=460 exceptions mentioned.