## The Annals of Probability

- Ann. Probab.
- Volume 18, Number 3 (1990), 1269-1283.

### 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.

#### Article information

**Source**

Ann. Probab., Volume 18, Number 3 (1990), 1269-1283.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176990746

**Digital Object Identifier**

doi:10.1214/aop/1176990746

**Mathematical Reviews number (MathSciNet)**

MR1062069

**Zentralblatt MATH identifier**

0713.62021

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62E15: Exact distribution theory

Secondary: 62G15: Tolerance and confidence regions

**Keywords**

Brownian bridge empirical process Kolmogorov-Smirnov statistics

#### Citation

Massart, P. The Tight Constant in the Dvoretzky-Kiefer-Wolfowitz Inequality. Ann. Probab. 18 (1990), no. 3, 1269--1283. doi:10.1214/aop/1176990746. https://projecteuclid.org/euclid.aop/1176990746