A short proof of the Dvoretzky--Kiefer--Wolfowitz--Massart inequality

Abstract

The Dvoretzky--Kiefer--Wolfowitz--Massart inequality gives a sub-Gaussian tail bound on the supremum norm distance between the empirical distribution function of a random sample and its population counterpart. We provide a short proof of a result that improves the existing bound in two respects. First, our one-sided bound holds without any restrictions on the failure probability, thereby verifying a conjecture of Birnbaum and McCarty (1958). Second, it is local in the sense that it holds uniformly over sub-intervals of the real line with an error rate that adapts to the behaviour of the population distribution function on the interval.

Theorems & Definitions (24)

• Theorem 1
• Corollary 1
• Corollary 2
• Corollary 3
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof : Proof of Theorem \ref{['thm:main_result_localised_dkw_m_inequality']}