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In the theory of [[probability]] and [[statistics]], the '''Dvoretzky–Kiefer–Wolfowitz inequality''' predicts how close an [[empirical distribution function|empirically determined distribution function]] will be to the [[cumulative distribution function|distribution function]] from which the empirical samples are drawn. It is named after [[Aryeh Dvoretzky]], [[Jack Kiefer (mathematician)|Jack Kiefer]], and [[Jacob Wolfowitz]], who in 1956 proved<ref name="Dvoretzky">{{citation
| last1 = Dvoretzky
| first1 = A.
| authorlink1 = Aryeh Dvoretzky
| last2 = Kiefer
| first2 = J.
| authorlink2 = Jack Kiefer (mathematician)
| last3 = Wolfowitz
| first3 = J.
| authorlink3 = Jacob Wolfowitz
| title = Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator
| journal = [[Annals of Mathematical Statistics]]
| volume = 27
| issue = 3
| year = 1956
| pages = 642–669
| url = http://projecteuclid.org/euclid.aoms/1177728174
| mr = 0083864
| doi = 10.1214/aoms/1177728174}}</ref>
the inequality with a 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'' = 1, <ref name="Massart">{{citation
| last=Massart
| first = P.
| title = The tight constant in the Dvoretzky–Kiefer–Wolfowitz inequality
| journal = The Annals of Probability
| volume = 18
| issue = 3
| year = 1990
| pages = 1269–1283
| url = http://projecteuclid.org/euclid.aop/1176990746
| mr = 1062069
| doi=10.1214/aop/1176990746}}</ref> confirming a conjecture due to Birnbaum and McCarty.<ref>{{cite journal
| mr=0093874
| zbl = 0087.34002
| last = Birnbaum
| first = Z. W.
| last2 = McCarty
| first2 = R. C.
| title = A distribution-free upper confidence bound for Pr{Y<X}, based on independent samples of X and Y
| journal = Annals of Mathematical Statistics
| volume = 29
| year = 1958
| pages = 558–562
| doi = 10.1214/aoms/1177706631
| url = http://projecteuclid.org/euclid.aoms/1177706631}}</ref>

==The DKW inequality==
Given a natural number ''n'', let ''X''1, ''X''2, …, ''Xn'' be real-valued [[independent and identically distributed]] [[random variable]]s with [[cumulative distribution function|distribution function]] ''F''(·). Let ''Fn'' denote the associated [[empirical distribution function]] defined by
: <math>
F_n(x) = \frac1n \sum_{i=1}^n \mathbf{1}_{\{X_i\leq x\}},\qquad x\in\mathbb{R}.
</math>

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
: <math>
\Pr\Bigl(\sup_{x\in\mathbb R} \bigl(F_n(x) - F(x)\bigr) > \varepsilon \Bigr) \le e^{-2n\varepsilon^2}\qquad \text{for every }\varepsilon\geq\sqrt{\tfrac{1}{2n}\ln2},
</math>

which also implies a two-sided estimate
<ref name="Kosorok">{{citation
| last1 = Kosorok
| first1 = M.R.
| title = Introduction to Empirical Processes and Semiparametric Inference
| year = 2008
| chapter = Chapter 11: Additional Empirical Process Results
| page = 210
| isbn = 9780387749778
| publisher=Springer }}</ref>
: <math>
\Pr\Bigl(\sup_{x\in\mathbb R} |F_n(x) - F(x)| > \varepsilon \Bigr) \le 2e^{-2n\varepsilon^2}\qquad \text{for every }\varepsilon>0.
</math>

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 test|Kolmogorov–Smirnov statistic]]. The inequalities above follow from the case where ''F'' corresponds to be the [[uniform distribution (continuous)|uniform distribution]] on [0,1] in view of the fact<ref name="Shorack">
{{citation
| last1 = Shorack
| first1 = G.R.
| last2 = Wellner
| first2 = J.A.
| title = Empirical Processes with Applications to Statistics
| year = 1986 |isbn=0-471-86725-X
| publisher=Wiley }}
</ref>
that ''Fn'' has the same distributions as ''Gn''(''F'') where ''Gn'' is the empirical distribution of
''U''1, ''U''2, …, ''Un'' where these are independent and Uniform(0,1), and noting that
: <math>
\sup_{x\in\mathbb R} |F_n(x) - F(x)|\stackrel{d}{=} \sup_{x \in \mathbb R} | G_n (F(x)) - F(x) | \le \sup_{0 \le t \le 1} | G_n (t) -t | ,
</math>
with equality if and only if ''F'' is continuous.

==References==
<references/>

{{DEFAULTSORT:Dvoretzky-Kiefer-Wolfowitz inequality}}
[[Category:Asymptotic statistical theory]]
[[Category:Statistical inequalities]]
[[Category:Empirical process]]

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