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July, 1990 The Tight Constant in the Dvoretzky-Kiefer-Wolfowitz Inequality
P. Massart
Ann. Probab. 18(3): 1269-1283 (July, 1990). DOI: 10.1214/aop/1176990746

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.

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P. Massart. "The Tight Constant in the Dvoretzky-Kiefer-Wolfowitz Inequality." Ann. Probab. 18 (3) 1269 - 1283, July, 1990. https://doi.org/10.1214/aop/1176990746

Information

Published: July, 1990
First available in Project Euclid: 19 April 2007

zbMATH: 0713.62021
MathSciNet: MR1062069
Digital Object Identifier: 10.1214/aop/1176990746

Subjects:
Primary: 62E15
Secondary: 62G15

Rights: Copyright © 1990 Institute of Mathematical Statistics

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Vol.18 • No. 3 • July, 1990
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