Two-sample Dvoretzky–Kiefer–Wolfowitz inequalities

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 n sup x | ( F n − F ) ( x ) | , then there is a constant C such that for any M > 0 , Pr ( K n > M ) ≤ C exp ( − 2 M 2 ) . 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 ≥ n 0 for some C depending on n 0 , with C = 2 if and only if n 0 ≥ 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 = 2 m if 101 ≤ m ≤ 300 . We conjecture that the DKWM inequality holds for all pairs m ≤ n with the 457 + 3 = 460 exceptions mentioned.