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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
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| | last1 = Dvoretzky
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| | first1 = A.
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| | authorlink1 = Aryeh Dvoretzky
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| | last2 = Kiefer
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| | first2 = J.
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| | authorlink2 = Jack Kiefer (mathematician)
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| | last3 = Wolfowitz
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| | first3 = J.
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| | authorlink3 = Jacob Wolfowitz
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| | title = Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator
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| | journal = [[Annals of Mathematical Statistics]]
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| | volume = 27
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| | issue = 3
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| | year = 1956
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| | pages = 642–669
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| | url = http://projecteuclid.org/euclid.aoms/1177728174
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| | mr = 0083864
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| | doi = 10.1214/aoms/1177728174}}</ref>
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| 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
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| | last=Massart
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| | first = P.
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| | title = The tight constant in the Dvoretzky–Kiefer–Wolfowitz inequality
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| | journal = The Annals of Probability
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| | volume = 18
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| | issue = 3
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| | year = 1990
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| | pages = 1269–1283
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| | url = http://projecteuclid.org/euclid.aop/1176990746
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| | mr = 1062069
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| | doi=10.1214/aop/1176990746}}</ref> confirming a conjecture due to Birnbaum and McCarty.<ref>{{cite journal
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| | mr=0093874
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| | zbl = 0087.34002
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| | last = Birnbaum
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| | first = Z. W.
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| | last2 = McCarty
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| | first2 = R. C.
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| | title = A distribution-free upper confidence bound for Pr{Y<X}, based on independent samples of X and Y
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| | journal = Annals of Mathematical Statistics
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| | volume = 29
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| | year = 1958
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| | pages = 558–562
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| | doi = 10.1214/aoms/1177706631
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| | url = http://projecteuclid.org/euclid.aoms/1177706631}}</ref>
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| ==The DKW inequality==
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| Given a natural number ''n'', let ''X''<sub>1</sub>, ''X''<sub>2</sub>, …, ''X<sub>n</sub>'' be real-valued [[independent and identically distributed]] [[random variable]]s with [[cumulative distribution function|distribution function]] ''F''(·). Let ''F<sub>n</sub>'' denote the associated [[empirical distribution function]] defined by
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| : <math> | |
| F_n(x) = \frac1n \sum_{i=1}^n \mathbf{1}_{\{X_i\leq x\}},\qquad x\in\mathbb{R}.
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| </math>
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| The Dvoretzky–Kiefer–Wolfowitz inequality bounds the probability that the [[random function]] ''F<sub>n</sub>'' differs from ''F'' by more than a given constant ''ε'' > 0 anywhere on the real line. More precisely, there is the one-sided estimate
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| : <math>
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| \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},
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| </math>
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| which also implies a two-sided estimate
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| <ref name="Kosorok">{{citation
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| | last1 = Kosorok
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| | first1 = M.R.
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| | title = Introduction to Empirical Processes and Semiparametric Inference
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| | year = 2008
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| | chapter = Chapter 11: Additional Empirical Process Results
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| | page = 210
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| | isbn = 9780387749778
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| | publisher=Springer }}</ref>
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| : <math>
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| \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.
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| </math>
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| 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">
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| {{citation
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| | last1 = Shorack
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| | first1 = G.R.
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| | last2 = Wellner
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| | first2 = J.A.
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| | title = Empirical Processes with Applications to Statistics
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| | year = 1986 |isbn=0-471-86725-X
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| | publisher=Wiley }}
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| </ref>
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| that ''F<sub>n</sub>'' has the same distributions as ''G<sub>n</sub>''(''F'') where ''G<sub>n</sub>'' is the empirical distribution of
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| ''U''<sub>1</sub>, ''U''<sub>2</sub>, …, ''U<sub>n</sub>'' where these are independent and Uniform(0,1), and noting that
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| : <math>
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| \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 | ,
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| </math>
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| with equality if and only if ''F'' is continuous.
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| ==References==
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| <references/>
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| {{DEFAULTSORT:Dvoretzky-Kiefer-Wolfowitz inequality}}
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| [[Category:Asymptotic statistical theory]]
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| [[Category:Statistical inequalities]]
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| [[Category:Empirical process]]
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