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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Empirical process ： ウィキペディア英語版 Empirical process In probability theory, an empirical process is a stochastic process that describes the proportion of objects in a system in a given state. For a process in a discrete state space a population continuous time Markov chain or Markov population model is a process which counts the number of objects in a given state (without rescaling). In mean field theory, limit theorems (as the number of objects becomes large) are considered and generalise the central limit theorem for empirical measures. Applications of the theory of empirical processes arise in non-parametric statistics. ==Definition== For ''X''1, ''X''2, ... ''X''''n'' independent and identically-distributed random variables in R with common cumulative distribution function ''F''(''x''), the empirical distribution function is defined by :$F\_n(x)=\backslash frac\backslash sum\_^n\; I\_(X\_i),$ where I''C'' is the indicator function of the set ''C''. For every (fixed) ''x'', ''F''''n''(''x'') is a sequence of random variables which converge to ''F''(''x'') almost surely by the strong law of large numbers. That is, ''F''''n'' converges to ''F'' pointwise. Glivenko and Cantelli strengthened this result by proving uniform convergence of ''F''''n'' to ''F'' by the Glivenko–Cantelli theorem. A centered and scaled version of the empirical measure is the signed measure :$G\_n(A)=\backslash sqrt(P\_n(A)-P(A))$ It induces a map on measurable functions ''f'' given by :$f\backslash mapsto\; G\_n\; f=\backslash sqrt(P\_n-P)f=\backslash sqrt\backslash left(\backslash frac\backslash sum\_^n\; f(X\_i)-\backslash mathbbf\backslash right)$ By the central limit theorem, $G\_n(A)$ converges in distribution to a normal random variable ''N''(0, ''P''(''A'')(1 − ''P''(''A''))) for fixed measurable set ''A''. Similarly, for a fixed function ''f'', $G\_nf$ converges in distribution to a normal random variable $N(0,\backslash mathbb(f-\backslash mathbbf)^2)$, provided that $\backslash mathbbf$ and $\backslash mathbbf^2$ exist. Definition :$\backslash bigl(G\_n(c)\backslash bigr)\_$, a collection of measurable subsets of ''S''. :$\backslash bigl(G\_nf\backslash bigr)\_$, a collection of measurable functions from ''S'' to $\backslash mathbb$. A significant result in the area of empirical processes is Donsker's theorem. It has led to a study of Donsker classes: sets of functions with the useful property that empirical processes indexed by these classes converge weakly to a certain Gaussian process. While it can be shown that Donsker classes are Glivenko–Cantelli classes, the converse is not true in general. 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「Empirical process」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.