Dvoretzky–Kiefer–Wolfowitz inequality

The above chart show an example application of the DKW inequality in constructing confidence bounds (in purple) around an empirical distribution function (in light blue). In this random draw, the true CDF (orange) is entirely contained within the DKW bounds.

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, …, 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

$F_{n}(x)={\frac {1}{n}}\sum _{i=1}^{n}\mathbf {1} _{\{X_{i}\leq x\}},\qquad x\in \mathbb {R} .$

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

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

$\Pr {\Bigl (}\sup _{x\in \mathbb {R} }{\bigl (}F_{n}(x)-F(x){\bigr )}>\varepsilon {\Bigr )}\leq e^{-2n\varepsilon ^{2}}\qquad {\text{for every }}\varepsilon \geq {\sqrt {{\tfrac {1}{2n}}\ln 2}},$

which also implies a two-sided estimate

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

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 F_n has the same distributions as G_n(F) where G_n is the empirical distribution of U₁, U₂, …, U_n where these are independent and Uniform(0,1), and noting that

$\sup _{x\in \mathbb {R} }|F_{n}(x)-F(x)|{\stackrel {d}{=}}\sup _{x\in \mathbb {R} }|G_{n}(F(x))-F(x)|\leq \sup _{0\leq t\leq 1}|G_{n}(t)-t|,$

with equality if and only if F is continuous.

Building CDF bands

The DKW inequality is one method for generating CDF based confidence bounds and producing a confidence band. The purpose of this confidence interval is to contain the entire CDF at the specified confidence level, while alternative approaches attempt to only achieve the confidence level on each individual point which can allow for a tighter bound. The DKW bounds runs parallel to, and is equally above and below, the empirical CDF. The equally spaced confidence interval around the empirical CDF allows for different rates of violations across the support of the distribution. In particular, it is more common for a CDF to be outside of the CDF bound estimated using the Dvoretzky–Kiefer–Wolfowitz inequality near the median of the distribution than near the endpoints of the distribution.

The confidence interval is often specified as

$F_{n}(x)-\varepsilon \leq F(x)\leq F_{n}(x)+\varepsilon {\text{ where }}\varepsilon ={\sqrt {\frac {\ln {\frac {2}{\alpha }}}{2n}}}$

See also

• Concentration inequality – a summary of bounds on sets of random variables.