An Elementary Proof of the Dvoretzky--Kiefer--Wolfowitz--Massart Inequality

Abstract

The Dvoretzky--Kiefer--Wolfowitz--Massart inequality gives an upper bound on the probability that the empirical distribution function of a finite sequence of independent random variables deviates from its theoretical value. It is widely used in statistics in tests and in the production of confidence intervals. The original proof by Massart, later reformulated by Dudley, is very long and technical. Recently, Reeve slightly extends Dvoretzky--Kiefer--Wolfowitz--Massart inequality and presents an alternative, much shorter proof, that uses a continuous time martingale. We present a simpler and less technical proof of the original Dvoretzky--Kiefer--Wolfowitz--Massart inequality, based on a discrete-time martingale and Sion's minimax theorem.