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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|线性回归代写linear regression代考|Other Model Violations

Without loss of generality, $E(e)=0$ for the unimodal MLR model with a constant, in that if $E(\tilde{e})=\mu \neq 0$, then the MLR model can always be written as $Y=\boldsymbol{x}^T \boldsymbol{\beta}+e$ where $E(e)=0$ and $E(Y) \equiv E(Y \mid \boldsymbol{x})=\boldsymbol{x}^T \boldsymbol{\beta}$. To see this claim notice that
$$\begin{gathered} Y=\tilde{\beta}1+x_2 \beta_2+\cdots+x_p \beta_p+\tilde{e}=\tilde{\beta}_1+E(\tilde{e})+x_2 \beta_2+\cdots+x_p \beta_p+\tilde{e}-E(\tilde{e}) \ =\beta_1+x_2 \beta_2+\cdots+x\mu \beta_p+e \end{gathered}$$
where $\beta_1=\tilde{\beta}_1+E(\tilde{e})$ and $e=\tilde{e}-E(\tilde{e})$. For example, if the errors $\tilde{e}_i$ are iid exponential $(\lambda)$ with $E\left(\tilde{e}_i\right)=\lambda$, use $e_i=\tilde{e}_i-\lambda$.

For least squares, it is crucial that $\sigma^2$ exists. For example, if the $e_i$ are iid Cauchy $(0,1)$, then $\sigma^2$ does not exist and the least squares estimators tend to perform very poorly.

The performance of least squares is analogous to the performance of $\bar{Y}$. The sample mean $\bar{Y}$ is a very good estimator of the population mean $\mu$ if the $Y_i$ are iid $N\left(\mu, \sigma^2\right)$, and $\bar{Y}$ is a good estimator of $\mu$ if the sample size is large and the $Y_i$ are iid with mean $\mu$ and variance $\sigma^2$. This result follows from the central limit theorem (CLT), but how “large is large” depends on the underlying distribution. The $n>30$ rule tends to hold for distributions that are close to normal in that they take on many values and $\sigma^2$ is not huge. Error distributions that are highly nonnormal with tiny $\sigma^2$ often need $n>>30$. For example, if $Y_1, \ldots, Y_n$ are iid Gamma $(1 / m, 1)$, then $n>25 m$ may be needed. Another example is distributions that take on one value with very high probability, e.g. a Poisson random variable with very small variance. Bimodal and multimodal distributions and highly skewed distributions with suggest using $n>5000$ for moderately skewed distributions.

## 统计代写|线性回归代写linear regression代考|The ANOVA F Test

After fitting least squares and checking the response and residual plots to see that an MLR model is reasonable, the next step is to check whether there is an MLR relationship between $Y$ and the nontrivial predictors $x_2, \ldots, x_p$. If at least one of these predictors is useful, then the OLS fitted values $\hat{Y}_i$ should be used. If none of the nontrivial predictors is useful, then $\bar{Y}$ will give as good predictions as $\hat{Y}_i$. Here the sample mean

$$\bar{Y}=\frac{1}{n} \sum_{i=1}^n Y_i$$
In the definition below, SSE is the sum of squared residuals and a residual $r_i=\hat{e}i=$ “errorhat.” In the literature “errorhat” is often rather misleadingly abbreviated as “error.” Definition 2.14. Assume that a constant is in the MLR model. a) The total sum of squares $$\text { SSTO }=\sum{i=1}^n\left(Y_i-\bar{Y}\right)^2 .$$
b) The regression sum of squares
$$S S R=\sum_{i=1}^n\left(\hat{Y}i-\bar{Y}\right)^2 .$$ c) The residual sum of squares or error sum of squares is $$S S E=\sum{i=1}^n\left(Y_i-\hat{Y}i\right)^2=\sum{i=1}^n r_i^2 .$$
The result in the following proposition is a property of least squares (OLS), not of the underlying MLR model. An obvious application is that given any two of SSTO, SSE, and SSR, the 3rd sum of squares can be found using the formula $S S T O=S S E+S S R$.

# 线性回归代写

## 统计代写|线性回归代写linear regression代考|Other Model Violations

$$Y=\tilde{\beta} 1+x_2 \beta_2+\cdots+x_p \beta_p+\tilde{e}=\tilde{\beta}_1+E(\tilde{e})+x_2 \beta_2+\cdots+x_p \beta_p+\tilde{e}-E(\tilde{e})=\beta_1+x_2 \beta_2+\cdots+x \mu \beta_p+e$$

## 统计代写|线性回归代写linear regression代考|The ANOVA F Test

$$\mathrm{SSTO}=\sum i=1^n\left(Y_i-\bar{Y}\right)^2 .$$
b) 回归平方和
$$S S R=\sum_{i=1}^n(\hat{Y} i-\bar{Y})^2 .$$
c) 残差平方和或误差平方和为
$$S S E=\sum i=1^n\left(Y_i-\hat{Y} i\right)^2=\sum i=1^n r_i^2 .$$

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## 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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师