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

2022年10月13日

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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

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