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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 Multivariate probit model ： ウィキペディア英語版 Multivariate probit model In statistics and econometrics, the multivariate probit model is a generalization of the probit model used to estimate several correlated binary outcomes jointly. For example, if it is believed that the decisions of sending at least one child to public school and that of voting in favor of a school budget are correlated (both decisions are binary), then the multivariate probit model would be appropriate for jointly predicting these two choices on an individual-specific basis. ==Example: bivariate probit== In the ordinary probit model, there is only one binary dependent variable $Y$ and so only one latent variable $Y^$ * is used. In contrast, in the bivariate probit model there are two binary dependent variables $Y\_1$ and $Y\_2$, so there are two latent variables: $Y^$ *_1 and $Y^$ *_2 . It is assumed that each observed variable takes on the value 1 if and only if its underlying continuous latent variable takes on a positive value: : $$ Y_1 = \begin 1 & \textY^ *_1>0, \\ 0 & \text, \end : $$ Y_2 = \begin 1 & \textY^ *_2>0, \\ 0 & \text, \end with : $$ \begin Y_1^ * = X_1\beta_1+\varepsilon_1 \\ Y_2^ * = X_2\beta_2+\varepsilon_2 \end and : $$ \begin \varepsilon_1\\ \varepsilon_2 \end \mid X \sim \mathcal \left( \begin 0\\ 0 \end, \begin 1&\rho\\ \rho&1 \end \right) Fitting the bivariate probit model involves estimating the values of $\backslash beta\_1,\backslash \; \backslash beta\_2,$ and $\backslash rho$. To do so, the likelihood of the model has to be maximized. This likelihood is : $$ \begin L(\beta_1,\beta_2) =\Big( \prod & P(Y_1=1,Y_2=1\mid\beta_1,\beta_2)^ P(Y_1=0,Y_2=1\mid\beta_1,\beta_2)^ \\() & \Big) \end Substituting the latent variables $Y\_1^$ * and $Y\_2^$ * in the probability functions and taking logs gives : $$ \begin \sum & \Big( Y_1Y_2 \ln P(\varepsilon_1>-X_1\beta_1,\varepsilon_2>-X_2\beta_2) \\() & \quad+(1-Y_1)(1-Y_2)\ln P(\varepsilon_1<-X_1\beta_1,\varepsilon_2<-X_2\beta_2) \Big). \end After some rewriting, the log-likelihood function becomes: : $$ \begin \sum & \Big ( Y_1Y_2\ln \Phi(X_1\beta_1,X_2\beta_2,\rho) \\() & \quad +(1-Y_1)(1-Y_2)\ln \Phi(-X_1\beta_1,-X_2\beta_2,\rho) \Big). \end Note that $\backslash Phi$ is the cumulative distribution function of the bivariate normal distribution. $Y\_1$ and $Y\_2$ in the log-likelihood function are observed variables being equal to one or zero. 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「Multivariate probit model」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.