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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'' = 1, confirming a conjecture due to Birnbaum and McCarty. ==The DKW inequality== Given a natural number ''n'', let ''X''1, ''X''2, …, ''Xn'' be real-valued independent and identically distributed random variables with distribution function ''F''(·). Let ''Fn'' denote the associated empirical distribution function defined by : 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 : which also implies a two-sided estimate : 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 () in view of the fact〔 〕 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 : with equality if and only if ''F'' is continuous. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Dvoretzky–Kiefer–Wolfowitz inequality」の詳細全文を読む スポンサード リンク
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