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

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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。