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
\Prl(\supx\inR|Fn(x)-F(x)|>\varepsilonr)\le
| -2n\varepsilon2 | |
| Ce |
forevery\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.[1] In 2021, Michael Naaman proved the multivariate version of the DKW inequality and generalized Massart's tightness result to the multivariate case, which results in a sharp constant of twice the dimension k of the space in which the observations are found: C = 2k.[2]
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
Fn(x)=
| 1n | |
| \sum |
| n | |
| i=1 |
| 1 | |
| \{Xi\leqx\ |
F(x)
X
x
Fn(x)
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
\Prl(\supx\inRl(Fn(x)-F(x)r)>\varepsilonr)\le
| -2n\varepsilon2 | |
| e |
forevery\varepsilon\geq\sqrt{\tfrac{1}{2n}ln2},
which also implies a two-sided estimate
\Prl(\supx\inR|Fn(x)-F(x)|>\varepsilonr)\le
| -2n\varepsilon2 | |
| 2e |
forevery\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] as Fn has the same distributions as Gn(F) where Gn is the empirical distribution ofU1, U2, …, Un where these are independent and Uniform(0,1), and noting that
\supx\inR|Fn(x)-F(x)| \stackrel{d}{=} \supx|Gn(F(x))-F(x)|\le\sup0|Gn(t)-t|,
In the multivariate case, X1, X2, …, Xn is an i.i.d. sequence of k-dimensional vectors. If Fn is the multivariate empirical cdf, then
| \Prl(\sup | |
| t\inRk |
|Fn(t)-F(t)|>\varepsilonr)\le
| -2n\varepsilon2 | |
| (n+1)ke |
The Dvoretzky–Kiefer–Wolfowitz inequality is obtained for the Kaplan–Meier estimator which is a right-censored data analog of the empirical distribution function
\Prl(\sqrtn\supt\in[0,infty)|(1-G(t))(Fn(t)-F(t))|>\varepsilonr)\le2.5
| -2\varepsilon2+C\varepsilon | |
| e |
\varepsilon>0
C<infty
Fn
G
See also: CDF-based nonparametric confidence interval.
The Dvoretzky–Kiefer–Wolfowitz inequality is one method for generating CDF-based confidence bounds and producing a confidence band, which is sometimes called the Kolmogorov–Smirnov 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 DKW inequality near the median of the distribution than near the endpoints of the distribution.
The interval that contains the true CDF,
F(x)
1-\alpha
Fn(x)-\varepsilon\leF(x)\leFn(x)+\varepsilon where\varepsilon=\sqrt{
| ||||
| 2n |
which is also a special case of the asymptotic procedure for the multivariate case, whereby one uses the following critical value
| d(\alpha,k) | |
| \sqrtn |
=\sqrt{
| ||||
| 2n |