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广义线性模型（GLiM，或GLM）是John Nelder和Robert Wedderburn在1972年制定的一种高级统计建模技术。它是一个包含许多其他模型的总称，它允许响应变量y具有除正态分布以外的误差分布。
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统计代写|广义线性模型代写generalized linear model代考|Derivation of the gamma model
The base density function for the gamma distribution is
$$
f(y ; \mu, \phi)=\frac{1}{y \Gamma(1 / \phi)}\left(\frac{y}{\mu \phi}\right)^{1 / \phi} \exp \left(-\frac{y}{\mu \phi}\right)
$$
In exponential-family form, the above probability density appears as
$$
f(y ; \mu, \phi)=\exp \left{\frac{y / \mu-(-\ln \mu)}{-\phi}+\frac{1-\phi}{\phi} \ln y-\frac{\ln \phi}{\phi}-\ln \Gamma\left(\frac{1}{\phi}\right)\right}
$$
This equation provides us with the link and the cumulant given by
$$
\begin{aligned}
\theta & =1 / \mu \
b(\theta) & =-\ln (\mu)
\end{aligned}
$$
From this, we may derive the mean and the variance
$$
\begin{aligned}
b^{\prime}(\theta) & =\frac{\partial b}{\partial \mu} \frac{\partial \mu}{\partial \theta} \
& =\left(-\frac{1}{\mu}\right)\left(-\mu^2\right) \
& =\mu \
b^{\prime \prime}(\theta) & =\frac{\partial^2 b}{\partial \mu^2}\left(\frac{\partial \mu}{\partial \theta}\right)+\frac{\partial b}{\partial \mu} \frac{\partial^2 \mu}{\partial \theta^2} \
& =(1)\left(-\mu^2\right) \
& =-\mu^2
\end{aligned}
$$
The variance here is an ingredient of the variance of $y$, which is found using $b^{\prime \prime}(\theta) a(\phi)=-\mu^2(-\phi)=\phi \mu^2$.
统计代写|广义线性模型代写generalized linear model代考|Example: Reciprocal link
We now present a rather famous example of a reciprocal-linked gamma dataset. The example first gained notoriety in McCullagh and Nelder (1989) and later was given a full examination in Hilbe and Turlach (1995). The example deals with car insurance claims (claims . dta) and models average claims for damage to an owner’s car on the basis of the policy holder’s age group (PA 1-8), the vehicle age group (VA 1-4), and the car group (CG 1-4). A frequency weight is given, called number, which represents the number of identical covariate patterns related to a particular outcome.
The criterion of coefficient of variation constancy across cell groups is assumed and has been validated in previous studies. Again, the gamma model is robust to deviation from this criterion, but not so much that it should not be assessed. This is particularly the case with the canonical link model.
A schema of the model given the reciprocal link is
$$
\eta=\left(\beta_0+\beta_1 \mathrm{PA}+\beta_2 \mathrm{CG}+\beta_3 \mathrm{VA}\right)^{-1}
$$
The main-effects model is displayed below. The levels for the age group, vehicle age group, and car group are included in the model through automated production of indicator variables by using Stata’s glm command with factor variables.
广义线性模型代考
统计代写|广义线性模型代写generalized linear model代考|Derivation of the gamma model
分布的基密度函数为
$$
f(y ; \mu, \phi)=\frac{1}{y \Gamma(1 / \phi)}\left(\frac{y}{\mu \phi}\right)^{1 / \phi} \exp \left(-\frac{y}{\mu \phi}\right)
$$
在指数族形式下，上述概率密度为
$$
f(y ; \mu, \phi)=\exp \left{\frac{y / \mu-(-\ln \mu)}{-\phi}+\frac{1-\phi}{\phi} \ln y-\frac{\ln \phi}{\phi}-\ln \Gamma\left(\frac{1}{\phi}\right)\right}
$$
这个方程为我们提供了由
$$
\begin{aligned}
\theta & =1 / \mu \
b(\theta) & =-\ln (\mu)
\end{aligned}
$$
由此，我们可以推导出均值和方差
$$
\begin{aligned}
b^{\prime}(\theta) & =\frac{\partial b}{\partial \mu} \frac{\partial \mu}{\partial \theta} \
& =\left(-\frac{1}{\mu}\right)\left(-\mu^2\right) \
& =\mu \
b^{\prime \prime}(\theta) & =\frac{\partial^2 b}{\partial \mu^2}\left(\frac{\partial \mu}{\partial \theta}\right)+\frac{\partial b}{\partial \mu} \frac{\partial^2 \mu}{\partial \theta^2} \
& =(1)\left(-\mu^2\right) \
& =-\mu^2
\end{aligned}
$$
这里的方差是$y$方差的一个成分，它是使用$b^{\prime \prime}(\theta) a(\phi)=-\mu^2(-\phi)=\phi \mu^2$找到的。
统计代写|广义线性模型代写generalized linear model代考|Example: Reciprocal link
我们现在给出一个相当著名的互链接伽马数据集的例子。这个例子首先在McCullagh和Nelder(1989)中声名狼藉，后来在Hilbe和Turlach(1995)中得到了全面的检验。这个例子处理汽车保险索赔(索赔)。数据)和车型根据投保人的年龄组别(PA 1-8)、车辆年龄组别(VA 1-4)和汽车年龄组别(CG 1-4)对车主的汽车损失进行平均索赔。给出一个频率权重，称为number，它表示与特定结果相关的相同协变量模式的数量。
假设了细胞组间变异常数系数的判据，并在前人的研究中得到了验证。再一次，伽玛模型对于偏离这个标准是稳健的，但并没有到不应该对其进行评估的程度。规范链接模型尤其如此。
给定相互链接的模型的模式是
$$
\eta=\left(\beta_0+\beta_1 \mathrm{PA}+\beta_2 \mathrm{CG}+\beta_3 \mathrm{VA}\right)^{-1}
$$
主效果模型如下图所示。通过使用Stata的glm命令和因子变量自动生成指标变量，将年龄组、车辆年龄组和汽车组的水平包含在模型中。
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