Dvoretzky–Kiefer–Wolfowitz inequality

In the theory of probability and statistics, the Dvoretzky–Kiefer–Wolfowitz inequality predicts how close an empirically determined distribution function will be to the distribution function from which the empirical samples are drawn. It is named after Aryeh Dvoretzky, Jack Kiefer, and Jacob Wolfowitz, who in 1956 proved the inequality 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.

Given a natural number n, let X₁, X₂, …, X_n be real-valued independent and identically distributed random variables with cumulative distribution function F(·). Let F_n denote the associated empirical distribution function defined by

So ${\displaystyle F(x)}$ is the probability that a single random variable ${\displaystyle X}$ is smaller than ${\displaystyle x}$, and ${\displaystyle F_{n}(x)}$ is the average number of random variables that are smaller than ${\displaystyle x}$.

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