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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 Edgeworth series ： ウィキペディア英語版 Edgeworth series The Gram–Charlier A series (named in honor of Jørgen Pedersen Gram and Carl Charlier), and the Edgeworth series (named in honor of Francis Ysidro Edgeworth) are series that approximate a probability distribution in terms of its cumulants.〔Stuart, A., & Kendall, M. G. (1968). The advanced theory of statistics. Hafner Publishing Company.〕 The series are the same; but, the arrangement of terms (and thus the accuracy of truncating the series) differ.〔Kolassa, J. E. (2006). Series approximation methods in statistics (Vol. 88). Springer Science & Business Media.〕 ==Gram–Charlier A series== The key idea of these expansions is to write the characteristic function of the distribution whose probability density function is to be approximated in terms of the characteristic function of a distribution with known and suitable properties, and to recover through the inverse Fourier transform. We examine a continuous random variable. Let $f$ be the characteristic function of its distribution whose density function is , and $\backslash kappa\_r$ its cumulants. We expand in terms of a known distribution with probability density function , characteristic function , and cumulants $\backslash gamma\_r$. The density is generally chosen to be that of the normal distribution, but other choices are possible as well. By the definition of the cumulants, we have (see Wallace, 1958)〔Wallace, D. L. (1958). Asymptotic approximations to distributions. The Annals of Mathematical Statistics, 635-654.〕 :$f(t)=\; \backslash exp\backslash left()$ and :$\backslash psi(t)=\backslash exp\backslash left(),$ which gives the following formal identity: :$f(t)=\backslash exp\backslash left()\backslash psi(t)\backslash ,.$ By the properties of the Fourier transform, $(it)^r\backslash psi(t)$ is the Fourier transform of $(-1)^r()(-x)$, where is the differential operator with respect to . Thus, after changing $x$ with $-x$ on both sides of the equation, we find for the formal expansion :$F(x)\; =\; \backslash exp\backslash left(-\; \backslash gamma\_r)\backslash frac\backslash right)\backslash Psi(x)\backslash ,.$ If is chosen as the normal density with mean and variance as given by , that is, mean $\backslash mu\; =\; \backslash kappa\_1$ and variance $\backslash sigma^2\; =\; \backslash kappa\_2$, then the expansion becomes :$F(x)\; =\; \backslash exp\backslash left()\backslash frac\backslash exp\backslash left().$ since $\backslash gamma\_r=0$ for all >2 as higher cumulants of the normal distribution are 0. By expanding the exponential and collecting terms according to the order of the derivatives, we arrive at the Gram–Charlier A series. If we include only the first two correction terms to the normal distribution, we obtain :$F(x)\; \backslash approx\; \backslash frac\backslash exp\backslash left()\backslash left()\backslash ,,$ with $H\_3(x)=x^3-3x$ and $H\_4(x)=x^4\; -\; 6x^2\; +\; 3$ (these are Hermite polynomials). Note that this expression is not guaranteed to be positive, and is therefore not a valid probability distribution. The Gram–Charlier A series diverges in many cases of interest—it converges only if $F(x)$ falls off faster than $\backslash exp(-(x^2)/4)$ at infinity (Cramér 1957). When it does not converge, the series is also not a true asymptotic expansion, because it is not possible to estimate the error of the expansion. For this reason, the Edgeworth series (see next section) is generally preferred over the Gram–Charlier A series. 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「Edgeworth series」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.