The Tight Constant in the Dvoretzky-Kiefer-Wolfowitz Inequality

Abstract

Let $\hat{F}_n$ denote the empirical distribution function for a sample of $n$ i.i.d. random variables with distribution function $F$. In 1956 Dvoretzky, Kiefer and Wolfowitz proved that $P\big(\sqrt{n} \sup_x(\hat{F}_n(x) - F(x)) > \lambda\big) \leq C \exp(-2\lambda^2),$ where $C$ is some unspecified constant. We show that $C$ can be taken as 1 (as conjectured by Birnbaum and McCarty in 1958), provided that $\exp(-2\lambda^2) \leq \frac{1}{2}$. In particular, the two-sided inequality $P\big(\sqrt{n} \sup_x|\hat{F}_n(x) - F(x)| > \lambda\big) \leq 2 \exp(-2\lambda^2)$ holds without any restriction on $\lambda$. In the one-sided as well as in the two-sided case, the constants cannot be further improved.