Posted on Categories:Generalized linear model, 广义线性模型, 统计代写, 统计代考

# 统计代写|广义线性模型代写Generalized linear model代考|Starting values for Newton–Raphson

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## 统计代写|广义线性模型代写Generalized linear model代考|Starting values for Newton–Raphson

To implement an algorithm for obtaining estimates of $\boldsymbol{\beta}$, we must have an initial guess for the parameters. There is no global mechanism for good starting values, but there is a reasonable solution for obtaining starting values when there is a constant in the model.
If the model includes a constant, then a common practice is to find the estimates for a constant-only model. For ML, this is a part of the model of interest, and knowing the likelihood for a constant-only model then allows a likelihood-ratio test for the parameters of the model of interest.
Often, the ML estimate for the constant-only model may be found analytically. For example, in chapter 12 we introduce the Poisson model. That model has a log likelihood given by
$$\mathcal{L}=\sum_{i=1}^n\left{y_i\left(x_i \boldsymbol{\beta}\right)-\exp \left(x_i \boldsymbol{\beta}\right)-\ln \Gamma\left(y_i+1\right)\right}$$
If we assume that there is only a constant term in the model, then the log likelihood may be written
$$\mathcal{L}=\sum_{i=1}^n\left{y_i \beta_0-\exp \left(\beta_0\right)-\ln \Gamma\left(y_i+1\right)\right}$$

## 统计代写|广义线性模型代写Generalized linear model代考|IRLS (using the expected Hessian)

Here we discuss the estimation algorithm known as IRLS. We begin by rewriting the (usual) updating formula from the Taylor series expansion presented in $(3.24)$ as
$$\Delta \beta^{(r-1)}=-\left{\frac{\partial^2 \mathcal{L}}{\partial\left(\boldsymbol{\beta}^{(r-1)}\right)^T \partial \boldsymbol{\beta}^{(r-1)}}\right}^{-1} \frac{\partial \mathcal{L}}{\partial \boldsymbol{\beta}^{(r-1)}}$$
and we replace the calculation of the observed Hessian (second derivatives) with its expectation. This substitution is known as the method of Fisher scoring. Because we know that $E\left{\left(y_i-\mu_i\right)^2\right}=v\left(\mu_i\right) a(\phi)$, we may write
\begin{aligned} -E\left(\frac{\partial^2 \mathcal{L}}{\partial \beta_j \partial \beta_k}\right) & =E\left(\frac{\partial \mathcal{L}}{\partial \beta_j} \frac{\partial \mathcal{L}}{\partial \beta_k}\right) \ & =\sum_{i=1}^n\left(\frac{\partial \mu}{\partial \eta}\right)i^2 \frac{1}{v\left(\mu_i\right) a(\phi)} x{j i} x_{k i} \end{aligned}
Substituting ( $\underline{3.40)}$ and $(\underline{3.20})$ into $\left(\underline{3.38)}\right.$ and rearranging, we see that $\delta \boldsymbol{\beta}^{(r-1)}$ is the solution to
$$\left{\sum_{i=1}^n \frac{1}{v\left(\mu_i\right) a(\phi)}\left(\frac{\partial \mu}{\partial \eta}\right)i^2 x{j i} x_{k i}\right} \Delta \boldsymbol{\beta}^{(r-1)}=\sum_{i=1}^n \frac{y_i-\mu_i}{v\left(\mu_i\right) a(\phi)}\left(\frac{\partial \mu}{\partial \eta}\right)i x_i^T$$ Using the $(r-1)$ superscript to emphasize calculation with $\boldsymbol{\beta}^{(r-1)}$, we may refer to the linear predictor as $$\eta_i^{(r-1)}-\text { offset }_i=\sum{k=1}^p x_{k i} \beta_k^{(r-1)}$$

## 统计代写|广义线性模型代写Generalized linear model代考|Starting values for Newton–Raphson

$$\mathcal{L}=\sum_{i=1}^n\left{y_i\left(x_i \boldsymbol{\beta}\right)-\exp \left(x_i \boldsymbol{\beta}\right)-\ln \Gamma\left(y_i+1\right)\right}$$

$$\mathcal{L}=\sum_{i=1}^n\left{y_i \beta_0-\exp \left(\beta_0\right)-\ln \Gamma\left(y_i+1\right)\right}$$

## 统计代写|广义线性模型代写Generalized linear model代考|IRLS (using the expected Hessian)

$$\Delta \beta^{(r-1)}=-\left{\frac{\partial^2 \mathcal{L}}{\partial\left(\boldsymbol{\beta}^{(r-1)}\right)^T \partial \boldsymbol{\beta}^{(r-1)}}\right}^{-1} \frac{\partial \mathcal{L}}{\partial \boldsymbol{\beta}^{(r-1)}}$$

\begin{aligned} -E\left(\frac{\partial^2 \mathcal{L}}{\partial \beta_j \partial \beta_k}\right) & =E\left(\frac{\partial \mathcal{L}}{\partial \beta_j} \frac{\partial \mathcal{L}}{\partial \beta_k}\right) \ & =\sum_{i=1}^n\left(\frac{\partial \mu}{\partial \eta}\right)i^2 \frac{1}{v\left(\mu_i\right) a(\phi)} x{j i} x_{k i} \end{aligned}

$$\left{\sum_{i=1}^n \frac{1}{v\left(\mu_i\right) a(\phi)}\left(\frac{\partial \mu}{\partial \eta}\right)i^2 x{j i} x_{k i}\right} \Delta \boldsymbol{\beta}^{(r-1)}=\sum_{i=1}^n \frac{y_i-\mu_i}{v\left(\mu_i\right) a(\phi)}\left(\frac{\partial \mu}{\partial \eta}\right)i x_i^T$$使用$(r-1)$上标来强调$\boldsymbol{\beta}^{(r-1)}$的计算，我们可以将线性预测器称为 $$\eta_i^{(r-1)}-\text { offset }i=\sum{k=1}^p x{k i} \beta_k^{(r-1)}$$

## MATLAB代写

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