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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 Generalized Dirichlet distribution ： ウィキペディア英語版 Generalized Dirichlet distribution In statistics, the generalized Dirichlet distribution (GD) is a generalization of the Dirichlet distribution with a more general covariance structure and almost twice the number of parameters. Random variables with a GD distribution are neutral but not completely neutral.〔R. J. Connor and J. E. Mosiman 1969. ''Concepts of independence for proportions with a generalization of the Dirichlet distribution''. Journal of the American Statistical Association, volume 64, pp194--206〕 The density function of $p\_1,\backslash ldots,p\_$ is :$$ \left()^ p_k^ \prod_^\left() where we define $p\_k=\; 1-\; \backslash sum\_^p\_i$. Here $B(x,y)$ denotes the Beta function. This reduces to the standard Dirichlet distribution if $b\_=a\_i+b\_i$ for $2\backslash leqslant\; i\backslash leqslant\; k-1$ ($b\_0$ is arbitrary). For example, if ''k=4'', then the density function of $p\_1,p\_2,p\_3$ is :$$ \left()^ p_1^p_2^p_3^p_4^\left(p_2+p_3+p_4\right)^\left(p_3+p_4\right)^ where and $p\_4=1-p\_1-p\_2-p\_3$. Connor and Mosimann define the PDF as they did for the following reason. Define random variables $z\_1,\backslash ldots,z\_$ with $z\_1=p\_1,\; z\_2=p\_2/\backslash left(1-p\_1\backslash right),\; z\_3=p\_3/\backslash left(1-(p\_1+p\_2)\backslash right),\backslash ldots,z\_i\; =\; p\_i/\backslash left(1-p\_1+\backslash cdots+p\_\backslash right)$. Then $p\_1,\backslash ldots,p\_k$ have the generalized Dirichlet distribution as parametrized above, if the $z\_i$ are iid beta with parameters $a\_i,b\_i$, $i=1,\backslash ldots,k-1$. == Alternative form given by Wong == Wong 〔T.-T. Wong 1998. ''Generalized Dirichlet distribution in Bayesian analysis''. Applied Mathematics and Computation, volume 97, pp165-181〕 gives the slightly more concise form for $x\_1+\backslash cdots\; +x\_k\backslash leqslant\; 1$ :$$ \prod_^k\frac} where $\backslash gamma\_j=\backslash beta\_j-\backslash alpha\_-\backslash beta\_$ for $1\backslash leqslant\; j\backslash leqslant\; k-1$ and $\backslash gamma\_k=\backslash beta\_k-1$. Note that Wong defines a distribution over a $k$ dimensional space (implicitly defining $x\_=1-\backslash sum\_^kx\_i$) while Connor and Mosiman use a $k-1$ dimensional space with $x\_k=1-\backslash sum\_^x\_i$. ==General moment function== If $X=\backslash left(X\_1,\backslash ldots,X\_k\backslash right)\backslash sim\; GD\_k\backslash left(\backslash alpha\_1,\backslash ldots,\backslash alpha\_k;\backslash beta\_1,\backslash ldots,\backslash beta\_k\backslash right)$, then :$$ E\left(X_k^\right )= \prod_^k \frac where $\backslash delta\_j=r\_+r\_+\backslash cdots\; +r\_k$ for $j=1,2,\backslash cdots,k-1$ and $\backslash delta\_k=0$. Thus :$$ E\left(X_j\right)=\frac\prod_^\frac. 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「Generalized Dirichlet 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.