# 统计代写|线性回归代写linear regression代考|STAT6450

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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
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## 统计代写|线性回归代写linear regression代考|Simple Linear Regression

The simple linear regression (SLR) model is
$$Y_i=\beta_1+\beta_2 X_i+e_i=\alpha+\beta X_i+e_i$$
where the $e_i$ are iid with $E\left(e_i\right)=0$ and $\operatorname{VAR}\left(e_i\right)=\sigma^2$ for $i=1, \ldots, n$. The $Y_i$ and $e_i$ are random variables while the $X_i$ are treated as known constants. The parameters $\beta_1, \beta_2$, and $\sigma^2$ are unknown constants that need to be estimated. (If the $X_i$ are random variables, then the model is conditional on the $X_i$ ‘s provided that the errors $e_i$ are independent of the $X_i$. Hence the $X_i$ ‘s are still treated as constants.)

The SLR model is a special case of the MLR model with $p=2, x_{i, 1} \equiv 1$, and $x_{i, 2}=X_i$. The normal SLR model adds the assumption that the $e_i$ are iid $\mathrm{N}\left(0, \sigma^2\right)$. That is, the error distribution is normal with zero mean and constant variance $\sigma^2$. The response variable $Y$ is the variable that you want to predict while the predictor variable $X$ is the variable used to predict the response. For SLR, $E\left(Y_i\right)=\beta_1+\beta_2 X_i$ and the line $E(Y)=\beta_1+\beta_2 X$ is the regression function. $\operatorname{VAR}\left(Y_i\right)=\sigma^2$.

For SLR, the least squares estimators $\hat{\beta}1$ and $\hat{\beta}_2$ minimize the least squares criterion $Q\left(\eta_1, \eta_2\right)=\sum{i=1}^n\left(Y_i-\eta_1-\eta_2 X_i\right)^2$. For a fixed $\eta_1$ and $\eta_2$, $Q$ is the sum of the squared vertical deviations from the line $Y=\eta_1+\eta_2 X$.
The least squares (OLS) line is $\hat{Y}=\hat{\beta}1+\hat{\beta}_2 X$ where the slope $$\hat{\beta}_2 \equiv \hat{\beta}=\frac{\sum{i=1}^n\left(X_i-\bar{X}\right)\left(Y_i-\bar{Y}\right)}{\sum_{i=1}^n\left(X_i-\bar{X}\right)^2}$$ and the intercept $\hat{\beta}1 \equiv \hat{\alpha}=\bar{Y}-\hat{\beta}_2 \bar{X}$. By the chain rule, $$\frac{\partial Q}{\partial \eta_1}=-2 \sum{i=1}^n\left(Y_i-\eta_1-\eta_2 X_i\right)$$

and
$$\frac{\partial^2 Q}{\partial \eta_1^2}=2 n$$

## 统计代写|线性回归代写linear regression代考|The No Intercept MLR Model

The no intercept MLR model, also known as regression through the origin, is still $\boldsymbol{Y}=\boldsymbol{X} \boldsymbol{\beta}+\boldsymbol{e}$, but there is no intercept in the model, so $\boldsymbol{X}$ does not contain a column of ones 1 . Hence the intercept term $\beta_1=\beta_1(1)$ is replaced by $\beta_1 x_{i 1}$. Software gives output for this model if the “no intercept” or “intercept $=F$ ” option is selected. For the no intercept model, the assumption $E(\boldsymbol{e})=0$ is important, and this assumption is rather strong.

Many of the usual MLR results still hold: $\boldsymbol{\beta}{O L S}=\left(\boldsymbol{X}^T \boldsymbol{X}\right)^{-1} \boldsymbol{X}^T \boldsymbol{Y}$, the vector of predicted fitted values $\widehat{\boldsymbol{Y}}=\boldsymbol{X} \hat{\boldsymbol{\beta}}{O L S}=\boldsymbol{H} \boldsymbol{Y}$ where the hat matrix $\boldsymbol{H}=\boldsymbol{X}\left(\boldsymbol{X}^T \boldsymbol{X}\right)^{-1} \boldsymbol{X}^T$ provided the inverse exists, and the vector of residuals is $\boldsymbol{r}=\boldsymbol{Y}-\widehat{\boldsymbol{Y}}$. The response plot and residual plot are made in the same way and should be made before performing inference.

The main difference in the output is the ANOVA table. The ANOVA $F$ test in Section $2.4$ tests $H o: \beta_2=\cdots=\beta_p=0$. The test in this section tests Ho : $\beta_1=\cdots=\beta_p=0 \equiv H o: \boldsymbol{\beta}=\mathbf{0}$. The following definition and test follows Guttman (1982, p. 147) closely.

Definition 2.25. Assume that $\boldsymbol{Y}=\boldsymbol{X} \boldsymbol{\beta}+e$ where the $e_i$ are iid. Assume that it is desired to test $H o: \boldsymbol{\beta}=\mathbf{0}$ versus $H_A: \boldsymbol{\beta} \neq \mathbf{0}$.
a) The uncorrected total sum of squares
$$S S T=\sum_{i=1}^n Y_i^2 .$$
b) The model sum of squares
$$S S M=\sum_{i=1}^n \hat{Y}i^2 .$$ c) The residual sum of squares or error sum of squares is $$S S E=\sum{i=1}^n\left(Y_i-\hat{Y}i\right)^2=\sum{i=1}^n r_i^2 .$$
d) The degrees of freedom (df) for SSM is $p$, the $\mathrm{df}$ for SSE is $n-p$ and the $\mathrm{df}$ for SST is $n$. The mean squares are MSE $=\mathrm{SSE} /(n-p)$ and MSM $=$ $\mathrm{SSM} / p$

# 线性回归代写

## 统计代写|线性回归代写线性回归代考|简单线性回归

$$Y_i=\beta_1+\beta_2 X_i+e_i=\alpha+\beta X_i+e_i$$
，其中$e_i$是iid, $E\left(e_i\right)=0$和$\operatorname{VAR}\left(e_i\right)=\sigma^2$是$i=1, \ldots, n$。$Y_i$和$e_i$是随机变量，而$X_i$被视为已知常数。参数$\beta_1, \beta_2$和$\sigma^2$是需要估计的未知常数。(如果$X_i$是随机变量，那么模型是有条件的$X_i$，只要错误$e_i$是独立于$X_i$。因此$X_i$仍然被视为常量。)

$$\frac{\partial^2 Q}{\partial \eta_1^2}=2 n$$

## 统计代写|线性回归代写线性回归代考|无拦截MLR模型

a)未校正的总平方和
$$S S T=\sum_{i=1}^n Y_i^2 .$$
b)模型平方和
$$S S M=\sum_{i=1}^n \hat{Y}i^2 .$$ c)平方和的残差和或误差平方和为$$S S E=\sum{i=1}^n\left(Y_i-\hat{Y}i\right)^2=\sum{i=1}^n r_i^2 .$$
d) SSM的自由度(df)为$p$, SSE的$\mathrm{df}$为$n-p$, SST的$\mathrm{df}$为$n$。均方为MSE $=\mathrm{SSE} /(n-p)$和MSM $=$$\mathrm{SSM} / p$

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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。

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