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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 measure ： ウィキペディア英語版 empirical measure In probability theory, an empirical measure is a random measure arising from a particular realization of a (usually finite) sequence of random variables. The precise definition is found below. Empirical measures are relevant to mathematical statistics. The motivation for studying empirical measures is that it is often impossible to know the true underlying probability measure $P$. We collect observations $X\_1,\; X\_2,\; \backslash dots\; ,\; X\_n$ and compute relative frequencies. We can estimate $P$, or a related distribution function $F$ by means of the empirical measure or empirical distribution function, respectively. These are uniformly good estimates under certain conditions. Theorems in the area of empirical processes provide rates of this convergence. ==Definition== Let $X\_1,\; X\_2,\; \backslash dots$ be a sequence of independent identically distributed random variables with values in the state space ''S'' with probability measure ''P''. Definition :The ''empirical measure'' ''P''''n'' is defined for measurable subsets of ''S'' and given by ::$P\_n(A)\; =\; \backslash sum\_^n\; I\_A(X\_i)=\backslash frac\backslash sum\_^n\; \backslash delta\_(A)$ :where $I\_A$ is the indicator function and $\backslash delta\_X$ is the Dirac measure. For a fixed measurable set ''A'', ''nP''''n''(''A'') is a binomial random variable with mean ''nP''(''A'') and variance ''nP''(''A'')(1 − ''P''(''A'')). In particular, ''P''''n''(''A'') is an unbiased estimator of ''P''(''A''). Definition :$\backslash bigl(P\_n(c)\backslash bigr)\_$, a collection of measurable subsets of ''S''. To generalize this notion further, observe that the empirical measure $P\_n$ maps measurable functions $f:S\backslash to\; \backslash mathbb$ to their ''empirical mean'', :$f\backslash mapsto\; P\_n\; f=\backslash int\_S\; f\; \backslash ,\; dP\_n=\backslash frac\backslash sum\_^n\; f(X\_i)$ In particular, the empirical measure of ''A'' is simply the empirical mean of the indicator function, ''P''''n''(''A'') = ''P''''n'' ''I''''A''. For a fixed measurable function $f$, $P\_nf$ is a random variable with mean $\backslash mathbbf$ and variance $\backslash frac\backslash mathbb(f\; -\backslash mathbb\; f)^2$. By the strong law of large numbers, ''P''n(''A'') converges to ''P''(''A'') almost surely for fixed ''A''. Similarly $P\_nf$ converges to $\backslash mathbb\; f$ almost surely for a fixed measurable function $f$. The problem of uniform convergence of ''P''''n'' to ''P'' was open until Vapnik and Chervonenkis solved it in 1968. If the class $\backslash mathcal$ (or $\backslash mathcal$) is Glivenko–Cantelli with respect to ''P'' then ''P''n'' converges to ''P'' uniformly over $c\backslash in\backslash mathcal$ (or $f\backslash in\; \backslash mathcal$). In other words, with probability 1 we have :$\backslash |P\_n-P\backslash |\_\backslash mathcal=\backslash sup\_=\backslash sup\_f|\backslash to\; 0.$ 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 measure」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.