update_stat3006

Author: RosaXIONG

public/notes/stat/Statistical Computing/2. Solution to Nonlinear Equations.md

@@ -334,7 +334,7 @@ l(\lambda | y_1, \ldots, y_n) = \sum_{i=1}^{n} \left( y_i \log \lambda - \lambda
 
   - We have independent count data ${y_1, . . . , y_n}$. For each $Y_i$ , $Y_i$ follows $Poi(λ_i)$, where $log(λ_i) = α + βx_i$ , $α$ and $β$ are parameters and xi is the fixed covariate. The p.d.f (probability density function) of $y_i$ is $f(y_j | \alpha, \beta, x_i) = e^{-e^{(\alpha + \beta x_i)}} \frac{(e^{(\alpha + \beta x_i)})^{y_i}}{y_i!}$. It follows that the joint p.d.f. is
 
-    ​            $$f(y_1, y_2, \ldots, y_n | \alpha, \beta) = \prod\limits_{i=1}^{n} e^{-e^{(\alpha + \beta x_i)}} \frac{(e^{\alpha + \beta x_i})^{y_i}}{y_i!}.$$
+    ​         	   $f(y_1, y_2, \ldots, y_n | \alpha, \beta) = \prod\limits_{i=1}^{n} e^{-e^{(\alpha + \beta x_i)}} \frac{(e^{\alpha + \beta x_i})^{y_i}}{y_i!}.$ 
 
     $$l(\alpha, \beta) = \log f(y_1, y_2, \ldots, y_n | \alpha, \beta) = \sum\limits_{i=1}^{n} [-e^{\alpha + \beta x_i} +y_i(\alpha+\beta x_i)- \log y_i!] $$ 
 
@@ -350,7 +350,7 @@ l(\lambda | y_1, \ldots, y_n) = \sum_{i=1}^{n} \left( y_i \log \lambda - \lambda
 
     The Newton step is
 
-    $$\begin{pmatrix}  \alpha_{k+1} \\  \beta_{k+1}  \end{pmatrix} 
-    =  \begin{pmatrix}  \alpha_k \\  \beta_k  \end{pmatrix} -  \begin{pmatrix}  -\sum_{i=1}^{n} e^{\alpha_k + \beta_k x_i} & -\sum_{i=1}^{n} x_i e^{\alpha_k + \beta_k x_i} \\  -\sum_{i=1}^{n} x_i e^{\alpha_k + \beta_k x_i} & -\sum_{i=1}^{n} x_i^2 e^{\alpha_k + \beta_k x_i}  \end{pmatrix}^{-1}  \begin{pmatrix}  -\sum_{i=1}^{n} e^{\alpha_k + \beta_k x_i} + \sum_{i=1}^{n} y_i \\  -\sum_{i=1}^{n} x_i e^{\alpha_k + \beta_k x_i} + \sum_{i=1}^{n} x_i y_i  \end{pmatrix}$$ 
+    $\begin{pmatrix}  \alpha_{k+1} \\  \beta_{k+1}  \end{pmatrix} 
+    =  \begin{pmatrix}  \alpha_k \\  \beta_k  \end{pmatrix} -  \begin{pmatrix}  -\sum_{i=1}^{n} e^{\alpha_k + \beta_k x_i} & -\sum_{i=1}^{n} x_i e^{\alpha_k + \beta_k x_i} \\  -\sum_{i=1}^{n} x_i e^{\alpha_k + \beta_k x_i} & -\sum_{i=1}^{n} x_i^2 e^{\alpha_k + \beta_k x_i}  \end{pmatrix}^{-1}  \begin{pmatrix}  -\sum_{i=1}^{n} e^{\alpha_k + \beta_k x_i} + \sum_{i=1}^{n} y_i \\  -\sum_{i=1}^{n} x_i e^{\alpha_k + \beta_k x_i} + \sum_{i=1}^{n} x_i y_i  \end{pmatrix}$