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## 统计代写|广义线性模型代写Generalized linear model代考|Derivation of the Bernoulli model

The binary or Bernoulli response probability distribution is simplified from the binomial distribution. The binomial denominator $k$ is equal to 1 , and the outcomes $y$ are constrained to the values ${0,1}$. Again, $y$ may initially be represented in your dataset as 1 or 2 , or some other alternative set. If so, you must change your data to the above description.

However, some programs use the opposite default behavior: they use 0 to denote a success and 1 to denote a failure. Transferring data unchanged between Stata and a package that uses this other codification will result in fitted models with reversed signs on the estimated coefficients.
The Bernoulli response probability function is
$$f(y ; p)=p^y(1-p)^{1-y}$$
The binomial normalization (combination) term has disappeared, which makes the function comparatively simple.
In canonical (exponential) form, the Bernoulli distribution is written
$$f(y ; p)=\exp \left{y \ln \left(\frac{p}{1-p}\right)+\ln (1-p)\right}$$
Below you will find the various functions and relationships that are required to complete the Bernoulli algorithm in canonical form. Because the canonical form is commonly referred to as the logit model, these functions can be thought of as those of the binary logit or logistic regression algorithm.
\begin{aligned} \theta & =\ln \left(\frac{p}{1-p}\right)=\eta=g(\mu)=\ln \left(\frac{\mu}{1-\mu}\right) \ g^{-1}(\theta) & =\frac{1}{1+\exp (-\eta)}=\frac{\exp (\eta)}{1+\exp (\eta)} \ b(\theta) & =-\ln (1-p)=-\ln (1-\mu) \ b^{\prime}(\theta) & =p=\mu \ b^{\prime \prime}(\theta) & =p(1-p)=\mu(1-\mu) \ g^{\prime}(\mu) & =\frac{1}{\mu(1-\mu)} \end{aligned}
The Bernoulli log-likelihood and deviance functions are
\begin{aligned} \mathcal{L}(\mu ; y) & =\sum_{i=1}^n\left{y_i \ln \left(\frac{\mu_i}{1-\mu_i}\right)+\ln \left(1-\mu_i\right)\right} \ D & =2 \sum_{i=1}^n\left{y_i \ln \left(\frac{y_i}{\mu_i}\right)+\left(1-y_i\right) \ln \left(\frac{1-y_i}{1-\mu_i}\right)\right} \end{aligned}

## 统计代写|广义线性模型代写Generalized linear model代考|The binomial regression algorithm

The canonical binomial algorithm is commonly referred to as logistic or logit regression. Traditionally, binomial models have three commonly used links: logit, probit, and complementary log-log (clog-log). There are other links that we will discuss. However, statisticians typically refer to a GLM-based regression by its link function, hence the still-used reference to probit or clog-log regression. For the same reason, statisticians generally referred to the canonical form as logit regression. This terminology is still used.

Over time, some researchers began referring to logit regression as logistic regression. They made a distinction based on the type of predictors in the model. A logit model comprised factor variables. The logistic model, on the other hand, had at least one continuous variable as a predictor. Although this distinction has now been largely discarded, we still find reference to it in older sources. Logit and logistic refer to the same basic model.

In the previous section, we provided all the functions required to construct the binomial algorithm. Because this is the canonical form, it is also the algorithm for logistic regression. We first give the grouped-response form because it encompasses the simpler model.

## 统计代写|广义线性模型代写Generalized linear model代考|Derivation of the Bernoulli model

$$f(y ; p)=p^y(1-p)^{1-y}$$

$$f(y ; p)=\exp \left{y \ln \left(\frac{p}{1-p}\right)+\ln (1-p)\right}$$

\begin{aligned} \theta & =\ln \left(\frac{p}{1-p}\right)=\eta=g(\mu)=\ln \left(\frac{\mu}{1-\mu}\right) \ g^{-1}(\theta) & =\frac{1}{1+\exp (-\eta)}=\frac{\exp (\eta)}{1+\exp (\eta)} \ b(\theta) & =-\ln (1-p)=-\ln (1-\mu) \ b^{\prime}(\theta) & =p=\mu \ b^{\prime \prime}(\theta) & =p(1-p)=\mu(1-\mu) \ g^{\prime}(\mu) & =\frac{1}{\mu(1-\mu)} \end{aligned}

\begin{aligned} \mathcal{L}(\mu ; y) & =\sum_{i=1}^n\left{y_i \ln \left(\frac{\mu_i}{1-\mu_i}\right)+\ln \left(1-\mu_i\right)\right} \ D & =2 \sum_{i=1}^n\left{y_i \ln \left(\frac{y_i}{\mu_i}\right)+\left(1-y_i\right) \ln \left(\frac{1-y_i}{1-\mu_i}\right)\right} \end{aligned}

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 统计代写|广义线性模型代写Generalized linear model代考|Example: The canonical inverse Gaussian

As mentioned earlier, the inverse Gaussian has found limited use in certain reliability studies. In particular, it is most appropriate when modeling a nonnegative response having a high initial peak, rapid drop, and long right tail. If a discrete response has many different values together with the same properties, then the inverse Gaussian may be appropriate for this type of data as well. A variety of other shapes can also be modeled using inverse Gaussian regression.
We can create a synthetic inverse Gaussian dataset using tools available in Stata. This will allow us to observe some of the properties and relationships we have discussed thus far.
Shown below, we create a dataset of 25,000 cases with two random halfnormal predictors having coefficients of 0.5 and 0.25 , respectively, and a constant of 1 . We used Stata’s rigaussian ( ) function to generate outcomes with inverse Gaussian distributed conditional variances and specified mean. We also specified a shape parameter of the inverse Gaussian distribution of $1 / 0.25$. We set the random-number seed to allow re-creation of these results.
. clear
. set seed 12345
. set obs 25000
number of observations (_N) was 0, now 25,000

• generate $x 1=\operatorname{abs}(\operatorname{rnormal}())$
• generate $\times 2=\operatorname{abs}(\operatorname{rnormal}())$
. generate eta $=1+.5 * \times 1+.25 * \times 2$
• generate $\mathrm{mu}=1 / \operatorname{sqrt}($ eta $)$
• generate $x i g=$ rigaussian (mu, 1/0.25)

The log and identity links are the primary noncanonical links associated with the inverse Gaussian distribution. This is similar to the gamma model.
We usually do not refer to a link function with full ML. The log-likelihood function is simply adjusted such that the inverse link, parameterized in terms of $x \boldsymbol{\beta}$, is substituted for each instance of $\mu$ in the log-likelihood function. Hence, for the log-inverse Gaussian, we have
$$\mathcal{L}=\sum_{i=1}^n\left[\frac{y_i /\left{2 \exp \left(x_i \boldsymbol{\beta}\right)^2\right}-1 / \exp \left(x_i \boldsymbol{\beta}\right)}{-\sigma^2}+\frac{1}{-2 y_i \sigma^2}-\frac{1}{2} \ln \left(2 \pi y_i^3 \sigma^2\right)\right]$$
Using the IRLS approach, we can shape the canonical form of the inverse Gaussian to the log link by making the following substitutions:
$\begin{array}{lcc} & \text { Canonical } & \text { Substitution } \ \text { link } & -1 /\left(2 \mu^2\right) & \ln (\mu) \ \text { inverse link } & (-2 \eta)^{-1 / 2} & \exp (\eta) \ g^{\prime}(\mu) & 1 / \mu^3 & 1 / \mu\end{array}$
All other aspects of the algorithm remain the same.
If we wish to amend the basic IRLS algorithm so that the observed information matrix is used-hence, making it similar to ML output – then another adjustment must be made. This has to do with a modification of the weight function, $w$. The log-inverse Gaussian algorithm, implementing a modified Newton-Raphson approach, appears as

Listing 7.2: IRLS algorithm for log-inverse Gaussian models using OIM
$$\begin{array}{ll} 1 & \mu={y-\operatorname{mean}(y)} / 2 \ 2 & \eta=\ln (\mu) \ 3 & \text { WHILE (abs }(\Delta \text { Dev ) > tolerance) }{ \ 4 & \left.W=1 / \mu^3(1 / \mu)^2\right}=1 / \mu \ 5 & z=\eta+(y-\mu) \mu \ 6 & W_0=W+2(y-\mu) \mu^2 \end{array}$$
$\beta=\left(X^{\mathrm{T}} W_0 X\right)^{-1} X^{\mathrm{T}} W_0 z$
$\eta=X \beta$
$\mu=1 / \exp (\eta)$
oldDev $=$ Dev
Dev $=\sum(y-\mu)^2 /\left(y \sigma^2 \mu^2\right)$
$\Delta$ Dev $=$ Dev – OldDev
})
$14 \quad \chi^2=\sum(y-\mu)^2 / \mu^3$

## 统计代写|广义线性模型代写Generalized linear model代考|Example: The canonical inverse Gaussian

． 清楚
． 设置种子12345
． 设置obs 25000

． 生成eta $=1+.5 * \times 1+.25 * \times 2$

$$\mathcal{L}=\sum_{i=1}^n\left[\frac{y_i /\left{2 \exp \left(x_i \boldsymbol{\beta}\right)^2\right}-1 / \exp \left(x_i \boldsymbol{\beta}\right)}{-\sigma^2}+\frac{1}{-2 y_i \sigma^2}-\frac{1}{2} \ln \left(2 \pi y_i^3 \sigma^2\right)\right]$$

$\begin{array}{lcc} & \text { Canonical } & \text { Substitution } \ \text { link } & -1 /\left(2 \mu^2\right) & \ln (\mu) \ \text { inverse link } & (-2 \eta)^{-1 / 2} & \exp (\eta) \ g^{\prime}(\mu) & 1 / \mu^3 & 1 / \mu\end{array}$

$$\begin{array}{ll} 1 & \mu={y-\operatorname{mean}(y)} / 2 \ 2 & \eta=\ln (\mu) \ 3 & \text { WHILE (abs }(\Delta \text { Dev ) > tolerance) }{ \ 4 & \left.W=1 / \mu^3(1 / \mu)^2\right}=1 / \mu \ 5 & z=\eta+(y-\mu) \mu \ 6 & W_0=W+2(y-\mu) \mu^2 \end{array}$$
$\beta=\left(X^{\mathrm{T}} W_0 X\right)^{-1} X^{\mathrm{T}} W_0 z$
$\eta=X \beta$
$\mu=1 / \exp (\eta)$
oldDev $=$ Dev
Dev $=\sum(y-\mu)^2 /\left(y \sigma^2 \mu^2\right)$
$\Delta$ Dev $=$ Dev – OldDev
｝）
$14 \quad \chi^2=\sum(y-\mu)^2 / \mu^3$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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

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## 统计代写|广义线性模型代写Generalized linear model代考|Using the gamma model for survival analysis

We mentioned earlier that exponential regression may be modeled using a loglinked gamma regression. Using the default ML version of glm provides the necessary means by which the observed information matrix (OIM) is used to calculate standard errors. This is the same method used by typical ML implementations of exponential regression. See the documentation for Stata’s streg command in Stata Survival Analysis Reference Manual for a summary of the exponential regression model.

Why are the exponential regression results the same as the log-gamma model results? The similarity can be seen in the likelihood functions. The exponential probability distribution has the form
$$f\left(\frac{y}{\mu}\right)=\frac{1}{\mu} \exp \left(-\frac{y}{\mu}\right)$$
where $\mu$ is parameterized as
$$\mu=\exp (x \beta)$$
such that the function appears as
$$f\left(\frac{y}{x \boldsymbol{\beta}}\right)=\frac{1}{\exp (x \boldsymbol{\beta})} \exp \left{-\frac{y}{\exp (x \boldsymbol{\beta})}\right}$$
The exponential log likelihood is thus
$$\mathcal{L}(\boldsymbol{\beta} ; y)=\sum_{i=1}^n\left{-x_i \boldsymbol{\beta}-\frac{y_i}{\exp \left(x_i \boldsymbol{\beta}\right)}\right}$$

## 统计代写|广义线性模型代写Generalized linear model代考|Derivation of the inverse Gaussian model

The inverse Gaussian probability distribution is a continuous distribution having two parameters given by
$$f\left(y ; \mu, \sigma^2\right)=\frac{1}{\sqrt{2 \pi y^3 \sigma^2}} \exp \left{-\frac{(y-\mu)^2}{2(\mu \sigma)^2 y}\right}$$
In exponential form, the inverse Gaussian distribution is given by
\begin{aligned} f\left(y ; \mu, \sigma^2\right) & =\exp \left{-\frac{(y-\mu)^2}{2 y(\mu \sigma)^2}-\frac{1}{2} \ln \left(2 \pi y^3 \sigma^2\right)\right} \ & =\exp \left{\frac{y / \mu^2-2 / \mu}{-2 \sigma^2}+\frac{1 / y}{-2 \sigma^2}+\frac{\sigma^2}{-2 \sigma^2} \ln \left(2 \pi y^3 \sigma^2\right)\right} \end{aligned}
The log-likelihood function may be written in exponential-family form by dropping the exponential and its associated braces.
$$\mathcal{L}=\sum_{i=1}^n\left{\frac{y_i /\left(2 \mu_i^2\right)-1 / \mu_i}{-\sigma^2}+\frac{1}{-2 y_i \sigma^2}-\frac{1}{2} \ln \left(2 \pi y_i^3 \sigma^2\right)\right}$$
GLM theory provides that, in canonical form, the link and cumulant functions are
\begin{aligned} \theta & =\frac{1}{2 \mu^2}=\frac{1}{2} \mu^{-2} \ b(\theta) & =\frac{1}{\mu} \ a(\phi) & =-\sigma^2 \end{aligned}
The sign and coefficient value are typically dropped from the inverse Gaussian link function when inserted into the GLM algorithm. It is normally given the value of $1 / \mu^2$, and the inverse link function is normally given the value of $1 / \sqrt{\eta}$.

## 统计代写|广义线性模型代写Generalized linear model代考|Using the gamma model for survival analysis

$$f\left(\frac{y}{\mu}\right)=\frac{1}{\mu} \exp \left(-\frac{y}{\mu}\right)$$

$$\mu=\exp (x \beta)$$

$$f\left(\frac{y}{x \boldsymbol{\beta}}\right)=\frac{1}{\exp (x \boldsymbol{\beta})} \exp \left{-\frac{y}{\exp (x \boldsymbol{\beta})}\right}$$

$$\mathcal{L}(\boldsymbol{\beta} ; y)=\sum_{i=1}^n\left{-x_i \boldsymbol{\beta}-\frac{y_i}{\exp \left(x_i \boldsymbol{\beta}\right)}\right}$$

## 统计代写|广义线性模型代写Generalized linear model代考|Derivation of the inverse Gaussian model

$$f\left(y ; \mu, \sigma^2\right)=\frac{1}{\sqrt{2 \pi y^3 \sigma^2}} \exp \left{-\frac{(y-\mu)^2}{2(\mu \sigma)^2 y}\right}$$

\begin{aligned} f\left(y ; \mu, \sigma^2\right) & =\exp \left{-\frac{(y-\mu)^2}{2 y(\mu \sigma)^2}-\frac{1}{2} \ln \left(2 \pi y^3 \sigma^2\right)\right} \ & =\exp \left{\frac{y / \mu^2-2 / \mu}{-2 \sigma^2}+\frac{1 / y}{-2 \sigma^2}+\frac{\sigma^2}{-2 \sigma^2} \ln \left(2 \pi y^3 \sigma^2\right)\right} \end{aligned}

$$\mathcal{L}=\sum_{i=1}^n\left{\frac{y_i /\left(2 \mu_i^2\right)-1 / \mu_i}{-\sigma^2}+\frac{1}{-2 y_i \sigma^2}-\frac{1}{2} \ln \left(2 \pi y_i^3 \sigma^2\right)\right}$$
GLM理论认为，在规范形式下，链接函数和累积函数为
\begin{aligned} \theta & =\frac{1}{2 \mu^2}=\frac{1}{2} \mu^{-2} \ b(\theta) & =\frac{1}{\mu} \ a(\phi) & =-\sigma^2 \end{aligned}

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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## 统计代写|广义线性模型代写Generalized linear model代考|ML estimation

ML estimation of the canonical-link Gaussian model requires specifying the Gaussian log likelihood in terms of $x \boldsymbol{\beta}$. Parameterized in terms of $x \boldsymbol{\beta}$, the Gaussian density function is given by
$$f\left(y ; \boldsymbol{\beta}, \sigma^2\right)=\frac{1}{\sqrt{2 \pi \sigma^2}} \exp \left{-\frac{1}{2 \sigma^2}(y-x \boldsymbol{\beta})^2\right}$$
After we express the joint density in exponential family form, the log likelihood becomes
$$\mathcal{L}\left(\mu, \sigma^2 ; y\right)=\sum_{i=1}^n\left{\frac{y_i x_i \boldsymbol{\beta}-\left(x_i \boldsymbol{\beta}\right)^2 / 2}{\sigma^2}-\frac{y_i^2}{2 \sigma^2}-\frac{1}{2} \ln \left(2 \pi \sigma^2\right)\right}$$
This result implies that
\begin{aligned} \theta & =x \boldsymbol{\beta} \ b(\theta) & =(x \boldsymbol{\beta})^2 / 2=\theta^2 / 2 \ b^{\prime}(\theta) & =x \boldsymbol{\beta} \ b^{\prime \prime}(\theta) & =1 \ a(\phi) & =\sigma^2 \end{aligned}
The ML algorithm estimates $\sigma^2$ in addition to standard parameters and linear predictor estimates. The estimated value of $\sigma^2$ should be identical to the dispersion values, but ML provides a standard error and confidence interval. The estimates differ from the usual ols estimates, which specify estimators for the regression model without assuming normality. The oLs estimate of $\sigma^2$ uses $n-p$ in the denominator, whereas the ML and GLM estimates use $n$; the numerator is the sum of the squared residuals. This slight difference (which makes no difference asymptotically and makes no difference in the estimated regression coefficients) is most noticeable in the estimated standard errors for models based on small samples.

## 统计代写|广义线性模型代写Generalized linear model代考|GLM log-Gaussian models

An important and perhaps the foremost reason for using GLM as a framework for model construction is the ability to easily adjust models to fit particular response data situations. The canonical-link Gaussian model assumes a normally distributed response. Although the normal model is robust to moderate deviations from this assumption, it is nevertheless the case that many data situations are not amenable to or appropriate for normal models.
Unfortunately, many researchers have used the canonical-link Gaussian for data situations that do not meet the assumptions on which the Gaussian model is based. Until recently, few software packages allowed users to model data by means other than the normal model; granted, many researchers had little training in nonnormal modeling. Most popular software packages now have GLM capabilities or at least implement many of the most widely used GLM procedures, including logistic, probit, and Poisson regression.

The log-Gaussian model is based on the Gaussian distribution. It uses the log rather than the (canonical) identity link . The log link is generally used for response data that can take only positive values on the continuous scale, or values greater than 0 . The data must be such that nonpositive values are not only absent but also theoretically precluded.

Before GLM, researchers usually modeled positive-only data with the normal model. However, they first took the natural log of the response prior to modeling. In so doing, they explicitly acknowledged the need to normalize the response relative to the predictors, thus accommodating one of the assumptions of the Gaussian model. The problem with this method is one of interpretation. Fitted values, as well as parameter estimates, are in terms of the log response. This obstacle often proves to be inconvenient.

## 统计代写|广义线性模型代写Generalized linear model代考|ML estimation

$$f\left(y ; \boldsymbol{\beta}, \sigma^2\right)=\frac{1}{\sqrt{2 \pi \sigma^2}} \exp \left{-\frac{1}{2 \sigma^2}(y-x \boldsymbol{\beta})^2\right}$$

$$\mathcal{L}\left(\mu, \sigma^2 ; y\right)=\sum_{i=1}^n\left{\frac{y_i x_i \boldsymbol{\beta}-\left(x_i \boldsymbol{\beta}\right)^2 / 2}{\sigma^2}-\frac{y_i^2}{2 \sigma^2}-\frac{1}{2} \ln \left(2 \pi \sigma^2\right)\right}$$

\begin{aligned} \theta & =x \boldsymbol{\beta} \ b(\theta) & =(x \boldsymbol{\beta})^2 / 2=\theta^2 / 2 \ b^{\prime}(\theta) & =x \boldsymbol{\beta} \ b^{\prime \prime}(\theta) & =1 \ a(\phi) & =\sigma^2 \end{aligned}

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 统计代写|广义线性模型代写Generalized linear model代考|Criterion measures

In the following definitions,
\begin{aligned} p & =\text { number of predictors } \ n & =\text { number of observations } \ L\left(M_k\right) & =\text { likelihood for model } k \ \mathcal{L}\left(M_k\right) & =\text { log likelihood for model } k \ D\left(M_k\right) & =\text { deviance of model } k \ G^2\left(M_k\right) & =\text { likelihood-ratio test of model } k \end{aligned}
Next, we provide the formulas for two criterion measures useful for model comparison. They include terms based on the log likelihood along with a penalty term based on the number of parameters in the model. In this way, the criterion measures seek to balance our competing desires for finding the best model (in terms of maximizing the likelihood) with model parsimony (including only those terms that significantly contribute to the model).
We introduce the two main model selection criterion measures below; also see Hilbe (1994, 2011) for more considerations.
AIC
The Akaike ( $\underline{1973})$ information criterion may be used to compare competing nested or nonnested models. The information criterion is a measure of the information lost in using the associated model. The goal is to find the model that has the lowest loss of information. In that sense, lower values of the criterion are indicative of a preferable model. Furthermore, a difference of greater than 2 indicates a marked preference for the model with the smaller criterion measure. When the measure is defined without scaling by the sample size, marked preference is called when there is a difference greater than two. The (scaled) formula is given by
$$\mathrm{AIC}=-2 \mathcal{L}\left(M_k\right)+2 p$$

## 统计代写|广义线性模型代写Generalized linear model代考|The interpretation of $R_2$ in linear regression

One of the most prevalent model measures is $R^2$. This statistic is usually discussed in introductory linear regression along with various ad hoc rules on its interpretation. A fortunate student is taught that there are many ways to interpret this statistic and that these interpretations have been generalized to areas outside linear regression.
For the linear regression model, we can define the $R^2$ measure in the following ways:
\begin{aligned} n & =\text { number of observations } \ p & =\text { number of predictors } \ M_\alpha & =\text { model with only an intercept } \ M_\beta & =\text { model with intercept and predictors } \end{aligned}
Percentage variance explained
The most popular interpretation is the percentage variance explained, where it can be shown that the $R^2$ statistic is equal to the ratio of the variance of the fitted values and to the total variance of the fitted values.
\begin{aligned} \text { RSS } & =\text { residual sum of squares }=\sum_{i=1}^n\left(y_i-\widehat{y}i\right)^2 \ \text { TSS } & =\text { total sum of squares }=\sum{i=1}^n\left(y_i-\bar{y}\right)^2 \ R^2 & =\frac{\text { TSS }- \text { RSS }}{\text { TSS }}=1-\frac{\text { RSS }}{\text { TSS }}=1-\frac{\sum_{i=1}^n\left(y_i-\widehat{y}i\right)^2}{\sum{i=1}^n\left(y_i-\bar{y}\right)^2} \end{aligned}

## 统计代写|广义线性模型代写Generalized linear model代考|Criterion measures

\begin{aligned} p & =\text { number of predictors } \ n & =\text { number of observations } \ L\left(M_k\right) & =\text { likelihood for model } k \ \mathcal{L}\left(M_k\right) & =\text { log likelihood for model } k \ D\left(M_k\right) & =\text { deviance of model } k \ G^2\left(M_k\right) & =\text { likelihood-ratio test of model } k \end{aligned}

aic
Akaike ($\underline{1973})$)信息标准可用于比较相互竞争的嵌套模型或非嵌套模型。信息标准是对使用相关模型时丢失的信息的度量。目标是找到信息损失最小的模型。从这个意义上说，较低的标准值表示较好的模型。此外，大于2的差异表明对具有较小标准度量的模型有明显的偏好。当测量的定义没有按样本大小进行缩放时，当差异大于2时，称为标记偏好。(缩放后的)公式由
$$\mathrm{AIC}=-2 \mathcal{L}\left(M_k\right)+2 p$$

## 统计代写|广义线性模型代写Generalized linear model代考|The interpretation of $R_2$ in linear regression

\begin{aligned} n & =\text { number of observations } \ p & =\text { number of predictors } \ M_\alpha & =\text { model with only an intercept } \ M_\beta & =\text { model with intercept and predictors } \end{aligned}

\begin{aligned} \text { RSS } & =\text { residual sum of squares }=\sum_{i=1}^n\left(y_i-\widehat{y}i\right)^2 \ \text { TSS } & =\text { total sum of squares }=\sum{i=1}^n\left(y_i-\bar{y}\right)^2 \ R^2 & =\frac{\text { TSS }- \text { RSS }}{\text { TSS }}=1-\frac{\text { RSS }}{\text { TSS }}=1-\frac{\sum_{i=1}^n\left(y_i-\widehat{y}i\right)^2}{\sum{i=1}^n\left(y_i-\bar{y}\right)^2} \end{aligned}

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 统计代写|广义线性模型代写Generalized linear model代考|Overdispersion

Overdispersion is a phenomenon that occurs when fitting data to discrete distributions, such as the binomial, Poisson, or negative binomial distribution. If the estimated dispersion after fitting is not near the assumed value, then the data may be overdispersed if the value is greater than expected or underdispersed if the value is less than expected. Overdispersion is far more common. For overdispersion, the simplest remedy is to assume a multiplicative factor in the usual implied variance. As such, the resulting covariance matrix will be inflated through multiplication by the estimated scale parameter. Care should be exercised because an inflated, estimated dispersion parameter may result from model misspecification rather than overdispersion, indicating that the model should be assessed for appropriateness by the researcher. Smith and Heitjan (1993) discuss testing and adjusting for departures from the nominal dispersion, Breslow (1990) discusses Poisson regression as well as other quasilikelihood models, and $\operatorname{Cox}(\underline{1983})$ gives a general overview of overdispersion. This topic is discussed in chapter 11.
Hilbe (2009) devotes an entire chapter to binomial overdispersion, and Hilbe (2011) devotes a chapter to extradispersed count models. Ganio and Schafer (1992), Lambert and Roeder (1995), and Dean and Lawless (1989) discuss diagnostics for overdispersion in GLMs.
A score test effectively compares the residuals with their expectation under the model. A test for overdispersion of the Poisson model, which compares its variance with the variance of a negative binomial, is given by
$$T_1^2=\frac{\left[\sum_{i=1}^n\left{\left(y_i-\widehat{\mu}i\right)^2-\left(1-\widehat{h}_i\right) \widehat{\mu}_i\right}\right]^2}{2 \sum{i=1}^n \widehat{\mu}_i^2} \sim \chi_1^2$$

## 统计代写|广义线性模型代写Generalized linear model代考|Assessing the link function

We may wish to investigate whether the link function is appropriate. For example, in Poisson regression, we may wish to examine whether the usual loglink (multiplicative effects) is appropriate compared with an identity link (additive effects). For binomial regression, we may wish to compare the logit link (symmetric about one half) with the complementary log-log link (asymmetric about one half).

Pregibon (1980) advocates the comparison of two link functions by embedding them in a parametric family of link functions. The Box-Cox family of power transforms
$$g(\mu ; \lambda)=\frac{\mu^\lambda-1}{\lambda}$$
and yields the log-link at $\lim {\lambda \rightarrow 0} g(\mu ; \lambda)$ and the identity link at $\lambda=1$. Likewise, the family $$g(\mu ; \lambda)=\ln \left{\frac{(1-\mu)^{-\lambda}-1}{\lambda}\right}$$ gives the logit link at $\lambda=1$ and the complementary log-log link at $\lim {\lambda \rightarrow 0} g(\mu ; \lambda)$.

## 统计代写|广义线性模型代写Generalized linear model代考|Overdispersion

Hilbe(2009)用了整整一章来讨论二项过分散，Hilbe(2011)用了一章来讨论外分散计数模型。Ganio和Schafer (1992)， Lambert和Roeder (1995)， Dean和Lawless(1989)讨论了glm中过度分散的诊断。

$$T_1^2=\frac{\left[\sum_{i=1}^n\left{\left(y_i-\widehat{\mu}i\right)^2-\left(1-\widehat{h}_i\right) \widehat{\mu}_i\right}\right]^2}{2 \sum{i=1}^n \widehat{\mu}_i^2} \sim \chi_1^2$$

## 统计代写|广义线性模型代写Generalized linear model代考|Assessing the link function

Pregibon(1980)主张通过将两个链接函数嵌入一个参数链接函数族来比较它们。Box-Cox家族的权力转换
$$g(\mu ; \lambda)=\frac{\mu^\lambda-1}{\lambda}$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 统计代写|广义线性模型代写Generalized linear model代考|Modified sandwich

If observations are grouped because they are correlated (perhaps because the data are really panel data), then the sandwich estimate is calculated, where $n_i$ refers to the observations for each panel $i$ and $x_{i j}$ refers to the row of the matrix $X$ associated with the $j$ th observation for subject $i$, using
$$\widehat{B}{\mathrm{MS}}=\sum{i=1}^n\left{\sum_{j=1}^{n_i} x_{i j}^T \frac{y_{i j}-\widehat{\mu}{i j}}{v\left(\widehat{\mu}{i j}\right)}\left(\frac{\partial \mu}{\partial \eta}\right){i j} \widehat{\phi}\right}\left{\sum{j=1}^{n_i} \frac{y_{i j}-\widehat{\mu}{i j}}{v\left(\widehat{\mu}{i j}\right)}\left(\frac{\partial \mu}{\partial \eta}\right){i j} \widehat{\phi} x{i j}\right}$$
as the modified (summed or partial) scores.
In either case, the calculation of $\widehat{V}H$ is the same and the modified sandwich estimate of variance is then $$\widehat{V}{\mathrm{MS}}=\widehat{V}H^{-1} \widehat{B}{\mathrm{MS}} \widehat{V}_H^{-1}$$
The problem with referring to the sandwich estimate of variance as “robust” and the modified sandwich estimate of variance as “robust cluster” is that it implies that robust standard errors are bigger than usual (Hessian) standard errors and that robust cluster standard errors are bigger still. This is a false conclusion. See Carroll et al. (1998) for a lucid comparison of usual and robust standard errors. A comparison of the sandwich estimate of variance and the modified sandwich estimate of variance depends on the within-panel correlation of the score terms. If the within-panel correlation is negative, then the panel score sums of residuals will be small, and the panel score sums will have less variability than the variability of the individual scores. This will lead to the modified sandwich standard errors being smaller than the sandwich standard errors.

## 统计代写|广义线性模型代写Generalized linear model代考|Unbiased sandwich

The sandwich estimate of variance has been applied in many cases and is becoming more common in statistical software. One area of current research concerns the small-sample properties of this variance estimate. There are two main modifications to the sandwich estimate of variance in constructing confidence intervals. The first is a degrees-of-freedom correction (scale factor), and the second is the use of a more conservative distribution (“heavier-tailed” than the normal).
Acknowledging that the usual sandwich estimate is biased, we may calculate an unbiased sandwich estimate of variance with improved small-sample performance in coverage probability. This modification is a scale factor multiplier motivated by the knowledge that the variance of the estimated residuals are biased on terms of the $i$ th diagonal element of the hat matrix, $h_i$, defined in $(\underline{4.5})$ :
$$V\left(\widehat{\epsilon}_i\right)=\sigma^2\left(1-h_i\right)$$

We can adjust for the bias of the contribution from the scores, where $x_i$ refers to the $i$ th row of the matrix $X$, using
$$\widehat{B}{\mathrm{US}}=\sum{i=1}^n x_i^T\left{\frac{y_i-\widehat{\mu}i}{v\left(\widehat{\mu}_i\right)}\left(\frac{\partial \mu}{\partial \eta}\right)_i \widehat{\phi}\right}^2 \frac{x_i}{1-\widehat{h}_i}$$ where the (unbiased) sandwich estimate of variance is then $$\widehat{V}{\mathrm{US}}=\widehat{V}H^{-1} \widehat{B}{\mathrm{US}} \widehat{V}_H^{-1}$$

## 统计代写|广义线性模型代写Generalized linear model代考|Modified sandwich

$$\widehat{B}{\mathrm{MS}}=\sum{i=1}^n\left{\sum_{j=1}^{n_i} x_{i j}^T \frac{y_{i j}-\widehat{\mu}{i j}}{v\left(\widehat{\mu}{i j}\right)}\left(\frac{\partial \mu}{\partial \eta}\right){i j} \widehat{\phi}\right}\left{\sum{j=1}^{n_i} \frac{y_{i j}-\widehat{\mu}{i j}}{v\left(\widehat{\mu}{i j}\right)}\left(\frac{\partial \mu}{\partial \eta}\right){i j} \widehat{\phi} x{i j}\right}$$

## 统计代写|广义线性模型代写Generalized linear model代考|Unbiased sandwich

$$V\left(\widehat{\epsilon}_i\right)=\sigma^2\left(1-h_i\right)$$

$$\widehat{B}{\mathrm{US}}=\sum{i=1}^n x_i^T\left{\frac{y_i-\widehat{\mu}i}{v\left(\widehat{\mu}_i\right)}\left(\frac{\partial \mu}{\partial \eta}\right)_i \widehat{\phi}\right}^2 \frac{x_i}{1-\widehat{h}_i}$$其中(无偏的)夹心方差估计为 $$\widehat{V}{\mathrm{US}}=\widehat{V}H^{-1} \widehat{B}{\mathrm{US}} \widehat{V}_H^{-1}$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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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

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 统计代写|广义线性模型代写Generalized linear model代考|Components

Cited in various places such as Hilbe (1993b) and Francis, Green, and Payne (1993), GLMs are characterized by an expanded itemized list given by the following:
A random component for the response, $y$, which has the characteristic variance of a distribution that belongs to the exponential family.
A linear systematic component relating the linear predictor, $\eta=X \boldsymbol{\beta}$, to the product of the design matrix $X$ and the parameters $\boldsymbol{\beta}$.
A known monotonic, one-to-one, differentiable link function $g(\cdot)$ relating the linear predictor to the fitted values. Because the function is one-to-one, there is an inverse function relating the mean expected response, $E(y)=\mu$, to the linear predictor such that $\mu=g^{-1}(\eta)=E(y)$.
The variance may change with the covariates only as a function of the mean.
There is one IRLS algorithm that suffices to fit all members of the class.
Item 5 is of special interest. The traditional formulation of the theory certainly supposed that there was one algorithm that could fit all GLMs. We will see later how this was implemented. However, there have been extensions to this traditional viewpoint. Adjustments to the weight function have been added to match the usual Newton-Raphson algorithms more closely and so that more appropriate standard errors may be calculated for noncanonical link models. Such features as scaling and robust variance estimators have also been added to the basic algorithm. More importantly, sometimes a traditional GLM must be restructured and fit using a model-specific Newton-Raphson algorithm. Of course, one may simply define a GLM as a model requiring only the standard approach but doing so would severely limit the range of possible models. We prefer to think of a GLM as a model that is ultimately based on the probability function belonging to the exponential family of distributions, but with the proviso that this criterion may be relaxed to include quasilikelihood models as well as certain types of multinomial, truncated , censored , and inflated models. Most of the latter tvpe require a Newton-Raphson approach rather than the traditional IRLS algorithm.

## 统计代写|广义线性模型代写Generalized linear model代考|Assumptions

The link function relates the mean $\mu=E(y)$ to the linear predictor $X \boldsymbol{\beta}$, and the variance function relates the variance as a function of the mean $V(y)=a(\phi) v(\mu)$, where $a(\phi)$ is the scale factor. For the Poisson, binomial, and negative binomial variance models, $a(\phi)=1$.
Breslow (1996) points out that the critical assumptions in the GLM framework may be stated as follows:
Statistical independence of the $n$ observations.
The variance function $v(\mu)$ is correctly specified.
$a(\phi)$ is correctly specified (1 for Poisson, binomial, and negative binomial).
The link function is correctly specified.
Explanatory variables are of the correct form.
There is no undue influence of the individual observations on the fit.
As a simple illustration, in table 2.1 we demonstrate the effect of the assumed variance function on the model and fitted values of a simple GLM.
Note: The models are all fit using the identity link, and the data consist of three observations $(y, x)={(1,1),(2,2),(9,3)}$. The fitted models are included in the last column.

## 统计代写|广义线性模型代写Generalized linear model代考|Assumptions

Breslow(1996)指出，GLM框架中的关键假设可以表述如下:
$n$观测值的统计独立性。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## avatest™帮您通过考试

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## 统计代写|广义线性模型代写Generalized linear model代考|More Empirical Demonstration

To demonstrate the potential improvement of the new CMMP procedure, another simulation study was carried out. In Jiang et al. (2018), the authors showed that CMMP significantly outperforms the standard regression prediction (RP) method. On the other hand, the authors have not compared CMMP with mixed model prediction, such as the EBLUP (see Sect. 2.3.2), which is known to outperform RP as well. In the current simulation, we consider a case where there is no exact match between the new observation and a group in the training data, a situation that is practical. More specifically, the training data satisfy
$$y_{i j}=\beta_0+\beta_1 w_i+\alpha_i+\epsilon_{i j}$$
$i=1, \ldots, m, j=1, \ldots, n_i$, where $w_i$ is an observed, cluster-level covariate, $\alpha_i$ is a cluster-specific random effect, and $\epsilon_{i j}$ is an error. The random effects and errors are independent with $\alpha_i \sim N(0, G)$ and $\epsilon_{i j} \sim N(0, R)$. The new observation, on the other hand, satisfies
$$y_{\text {new }}=\beta_0+\beta_1 w_1+\alpha_1+\delta+\epsilon_{\text {new }}$$
where $\delta, \epsilon_{\text {new }}$ are independent with $\delta \sim N(0, D)$ and $\epsilon_{\text {new }} \sim N(0, R)$ and $\left(\delta, \epsilon_{\text {new }}\right)$ are independent with the training data. It is seen that, because of $\delta$, there is no exact match between the new random effect (which is $\alpha_1+\delta$ ) and one of the random effects $\alpha_i$ associated with the training data; however, the value of $D$ is small, $D=10^{-4}$; hence there is an approximate match between the new random effect and $\alpha_1$, the random effect associated with the first group in the training data.

We consider $m=50$. The $n_i$ are chosen according to one of the following four patterns:

1. $n_i=5,1 \leq i \leq m / 2 ; n_i=25, m / 2+1 \leq i \leq m$;
2. $n_i=50,1 \leq i \leq m / 2 ; n_i=250, m / 2+1 \leq i \leq m$;
3. $n_i=25,1 \leq i \leq m / 2 ; n_i=5, m / 2+1 \leq i \leq m$;
4. $n_i=250,1 \leq i \leq m / 2 ; n_i=50, m / 2+1 \leq i \leq m$.

## 统计代写|广义线性模型代写Generalized linear model代考|Prediction Interval

Prediction intervals are of substantial practical interest. Here, we follow the NER model (2.67), but with the additional assumption that the new error, $\epsilon_{\mathrm{n}, j}$ in (2.58), is distributed as $N(0, R)$, where $R$ is the same variance as that of $\epsilon_{i j}$ in (2.67). Still, it is not necessary to assume that $\alpha_{\text {new }}=\alpha_I$ has the same distribution, or even the same variance, as the $\alpha_i$ in (2.57). This would include both the matched and unmatched cases. Consider the following prediction interval for $\theta=x_{\mathrm{n}}^{\prime} \beta+\alpha_{\text {new }}$ :
$$\left[\hat{\theta}-z_{a / 2} \sqrt{\frac{\hat{R}}{n_{\text {new }}}}, \hat{\theta}+z_{a / 2} \sqrt{\frac{\hat{R}}{n_{\text {new }}}}\right] \text {, }$$
where $\hat{\theta}$ is the CMEP of $\theta, \hat{R}$ is the REML estimator of $R$, and $z_a$ is the critical value so that $\mathrm{P}\left(Z>z_a\right)=a$ for $Z \sim N(0,1)$. For a future observation, $y_{\mathrm{f}}$, we assume that it shares the same mixed effects as the observed new observations $y_{\mathrm{n}, j}, 1 \leq j \leq$ $n_{\text {new }}$ in $(2.58)$, that is,
$$y_{\mathrm{f}}=\theta+\epsilon_{\mathrm{f}}$$
where $\epsilon_{\mathrm{f}}$ is a new error that is distributed as $N(0, R)$ and independent with all of the $\alpha$ ‘s and other $\epsilon$ ‘s. Consider the following prediction interval for $y_{\mathrm{f}}$ :
$$\left[\hat{\theta}-z_{a / 2} \sqrt{\left(1+n_{\text {new }}^{-1}\right) \hat{R}}, \hat{\theta}+z_{a / 2} \sqrt{\left(1+n_{\text {new }}^{-1}\right) \hat{R}}\right],$$
where $\hat{\theta}, \hat{R}$ are the same as in (2.72). Under suitable conditions, it can be shown that the prediction intervals (2.72) and (2.74) have asymptotically the correct coverage probability. Furthermore, empirical results show that the CMMP-based prediction intervals are more accurate than the RP-based prediction intervals. See Jiang et al. (2018) for details.

## 计代写|广义线性模型代写Generalized linear model代考|More Empirical Demonstration

$$y_{i j}=\beta_0+\beta_1 w_i+\alpha_i+\epsilon_{i j}$$
$i=1, \ldots, m, j=1, \ldots, n_i$，其中$w_i$是观察到的集群级协变量，$\alpha_i$是特定于集群的随机效应，$\epsilon_{i j}$是一个误差。随机效应和误差与$\alpha_i \sim N(0, G)$和$\epsilon_{i j} \sim N(0, R)$无关。另一方面，新的观察结果满足了
$$y_{\text {new }}=\beta_0+\beta_1 w_1+\alpha_1+\delta+\epsilon_{\text {new }}$$

$n_i=5,1 \leq i \leq m / 2 ; n_i=25, m / 2+1 \leq i \leq m$；

$n_i=50,1 \leq i \leq m / 2 ; n_i=250, m / 2+1 \leq i \leq m$；

$n_i=25,1 \leq i \leq m / 2 ; n_i=5, m / 2+1 \leq i \leq m$；

$n_i=250,1 \leq i \leq m / 2 ; n_i=50, m / 2+1 \leq i \leq m$．

## 统计代写|广义线性模型代写Generalized linear model代考|Prediction Interval

$$\left[\hat{\theta}-z_{a / 2} \sqrt{\frac{\hat{R}}{n_{\text {new }}}}, \hat{\theta}+z_{a / 2} \sqrt{\frac{\hat{R}}{n_{\text {new }}}}\right] \text {, }$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。