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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 Scaled inverse chi-squared distribution ： ウィキペディア英語版 Scaled inverse chi-squared distribution ~ \frac\right )},\frac\right) \left/\Gamma\left(\frac\right)\right.| mean =$\backslash frac$ for $\backslash nu\; >2\backslash ,$| median =| mode =$\backslash frac$| variance =$\backslash frac$for $\backslash nu\; >4\backslash ,$| skewness =$\backslash frac\backslash sqrt$for $\backslash nu\; >6\backslash ,$| kurtosis =$\backslash frac$for $\backslash nu\; >8\backslash ,$| entropy =$\backslash frac$ \!+\!\ln\left(\frac\Gamma\left(\frac\right)\right) $\backslash !-\backslash !\backslash left(1\backslash !+\backslash !\backslash frac\backslash right)\backslash psi\backslash left(\backslash frac\backslash right)$| mgf =$\backslash frac)\}\backslash left(\backslash frac\backslash right)^\}\backslash !\backslash !K\_\}\backslash left(\backslash sqrt\backslash right)$| char =$\backslash frac)\}\backslash left(\backslash frac\backslash right)^\}\backslash !\backslash !K\_\}\backslash left(\backslash sqrt\backslash right)$| }} The scaled inverse chi-squared distribution is the distribution for ''x'' = 1/''s''2, where ''s''2 is a sample mean of the squares of ν independent normal random variables that have mean 0 and inverse variance 1/σ2 = τ2. The distribution is therefore parametrised by the two quantities ν and τ2, referred to as the ''number of chi-squared degrees of freedom'' and the ''scaling parameter'', respectively. This family of scaled inverse chi-squared distributions is closely related to two other distribution families, those of the inverse-chi-squared distribution and the inverse gamma distribution. Compared to the inverse-chi-squared distribution, the scaled distribution has an extra parameter ''τ''2, which scales the distribution horizontally and vertically, representing the inverse-variance of the original underlying process. Also, the scale inverse chi-squared distribution is presented as the distribution for the inverse of the ''mean'' of ν squared deviates, rather than the inverse of their ''sum''. The two distributions thus have the relation that if :$X\; \backslash sim\; \backslash mbox\backslash chi^2(\backslash nu,\; \backslash tau^2)$   then   $\backslash frac\; \backslash sim\; \backslash mbox\backslash chi^2(\backslash nu)$ Compared to the inverse gamma distribution, the scaled inverse chi-squared distribution describes the same data distribution, but using a different parametrization, which may be more convenient in some circumstances. Specifically, if :$X\; \backslash sim\; \backslash mbox\backslash chi^2(\backslash nu,\; \backslash tau^2)$   then   $X\; \backslash sim\; \backslash textrm\backslash left(\backslash frac,\; \backslash frac\backslash right)$ Either form may be used to represent the maximum entropy distribution for a fixed first inverse moment $(E(1/X))$ and first logarithmic moment $(E(\backslash ln(X))$. The scaled inverse chi-squared distribution also has a particular use in Bayesian statistics, somewhat unrelated to its use as a predictive distribution for ''x'' = 1/''s''2. Specifically, the scaled inverse chi-squared distribution can be used as a conjugate prior for the variance parameter of a normal distribution. In this context the scaling parameter is denoted by σ02 rather than by τ2, and has a different interpretation. The application has been more usually presented using the inverse gamma distribution formulation instead; however, some authors, following in particular Gelman ''et al.'' (1995/2004) argue that the inverse chi-squared parametrisation is more intuitive. ==Characterization== The probability density function of the scaled inverse chi-squared distribution extends over the domain $x>0$ and is :$$ f(x; \nu, \tau^2)= \frac~ \frac\right )},\frac\right) \left/\Gamma\left(\frac\right)\right. :$=Q\backslash left(\backslash frac,\backslash frac\backslash right)$ where $\backslash Gamma(a,x)$ is the incomplete Gamma function, $\backslash Gamma(x)$ is the Gamma function and $Q(a,x)$ is a regularized Gamma function. The characteristic function is :$\backslash varphi(t;\backslash nu,\backslash tau^2)=$ :$\backslash frac)\}\backslash left(\backslash frac\backslash right)^\}\backslash !\backslash !K\_\}\backslash left(\backslash sqrt\backslash right)\; ,$ where $K\_\}(z)$ is the modified Bessel function of the second kind. Differential equation $$ \left\} \left(\nu \tau ^2\right)^}\right)}\right\} 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「Scaled inverse chi-squared distribution」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.