Dvoretzky–Kiefer–Wolfowitz inequality

In the theory of probability and statistics, the Dvoretzky–Kiefer–Wolfowitz–Massart inequality (DKW inequality) provides a bound on the worst case distance of an empirically determined distribution function from its associated population distribution function. It is named after Aryeh Dvoretzky, Jack Kiefer, and Jacob Wolfowitz, who in 1956 proved the inequality

${\displaystyle \Pr {\Bigl (}\sup _{x\in \mathbb {R} }|F_{n}(x)-F(x)|>\varepsilon {\Bigr )}\leq Ce^{-2n\varepsilon ^{2}}\qquad {\text{for every }}\varepsilon >0.}$

with an unspecified multiplicative constant C in front of the exponent on the right-hand side.

In 1990, Pascal Massart proved the inequality with the sharp constant C = 2, confirming a conjecture due to Birnbaum and McCarty.