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Dvoretzky–Kiefer–Wolfowitz inequality

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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.

The DKW inequality

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

F n ( x ) = 1 n ∑ i = 1 n 1 { X i ≤ x } , x ∈ R .

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

The Dvoretzky–Kiefer–Wolfowitz inequality bounds the probability that the random function Fn differs from F by more than a given constant ε > 0 anywhere on the real line. More precisely, there is the one-sided estimate

Pr ( sup x ∈ R ( F n ( x ) − F ( x ) ) > ε ) ≤ e − 2 n ε 2 for every  ε ≥ 1 2 n ln ⁡ 2 ,

which also implies a two-sided estimate

Pr ( sup x ∈ R | F n ( x ) − F ( x ) | > ε ) ≤ 2 e − 2 n ε 2 for every  ε > 0.

This strengthens the Glivenko–Cantelli theorem by quantifying the rate of convergence as n tends to infinity. It also estimates the tail probability of the Kolmogorov–Smirnov statistic. The inequalities above follow from the case where F corresponds to be the uniform distribution on [0,1] in view of the fact that Fn has the same distributions as Gn(F) where Gn is the empirical distribution of U1, U2, …, Un where these are independent and Uniform(0,1), and noting that

sup x ∈ R | F n ( x ) − F ( x ) | = d sup x ∈ R | G n ( F ( x ) ) − F ( x ) | ≤ sup 0 ≤ t ≤ 1 | G n ( t ) − t | ,

with equality if and only if F is continuous.

References

Dvoretzky–Kiefer–Wolfowitz inequality Wikipedia
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