# 统计代写|线性回归分析代写linear regression analysis代考|STA4210

#### Doug I. Jones

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## 统计代写|线性回归分析代写linear regression analysis代考|Complements

The Least Squares Central Limit Theorem 2.8 is often a good approximation if $n \geq 10 p$ and the error distribution has “light tails,” i.e. the probability of an outlier is nearly 0 and the tails go to zero at an exponential rate or faster. For error distributions with heavier tails, much larger samples are needed, and the assumption that the variance $\sigma^2$ exists is crucial, e.g. Cauchy errors are not allowed. Norman and Streiner (1986, p. 63) recommend $n \geq 5 p$.
The classical MLR prediction interval does not work well and should be replaced by the Olive (2007) asymptotically optimal PI (2.20). Lei and Wasserman (2014) provide an alternative: use the Lei et al. (2013) PI $\left[\tilde{r}_L, \tilde{r}_U\right]$ on the residuals, then the PI for $Y_f$ is
$$\left[\hat{Y}_f+\tilde{r}_L, \hat{Y}_f+\tilde{r}_U\right]$$
Bootstrap PIs need more theory and instead of using $B=1000$ samples, use $B=\max (1000, n)$. See Olive (2014, pp. 279-285).

For the additive error regression model $Y=m(\boldsymbol{x})+e$, the response plot of $\hat{Y}=\hat{m}(\boldsymbol{x})$ vs. $Y$, with the identity line added as a visual aid, is used like the MLR response plot. We want $n \geq 10 d f$ where $d f$ is the degrees of freedom from fitting $\hat{m}$. Olive (2013a) provides PIs for this model, including the location model. These PIs are large sample PIs provided that the sample quantiles of the residuals are consistent estimators of the population quantiles of the errors. The response plot and PIs could also be used for methods described in James et al. (2013) such as ridge regression, lasso, principal components regression, and partial least squares. See Pelawa Watagoda and Olive (2017) if $n$ is not large compared to $p$.

## 统计代写|线性回归分析代写linear regression analysis代考|Lack of Fit Tests

Then $M S P E=S S P E /(n-c)$ is an unbiased estimator of $\sigma^2$ when model (2.29) holds, regardless of the form of $m$. The PE in SSPE stands for “pure error.”

Now SSLF $=S S E-S S P E=\sum_{j=1}^c n_j\left(\bar{Y}_j-\hat{Y}_j\right)^2$. Notice that $\bar{Y}_j$ is an unbiased estimator of $m\left(\boldsymbol{x}_j\right)$ while $\hat{Y}_j$ is an estimator of $m$ if the MLR model is appropriate: $m\left(\boldsymbol{x}_j\right)=\boldsymbol{x}_j^T \boldsymbol{\beta}$. Hence SSLF and MSLF can be very large if the MLR model is not appropriate.

The 4 step lack of fit test is i) Ho: no evidence of MLR lack of fit, $H_A$ : there is lack of fit for the MLR model.
ii) $F_{L F}=M S L F / M S P E$.
iii) The pval $=P\left(F_{c-p, n-c}>F_{L F}\right)$.
iv) Reject Ho if pval $\leq \delta$ and state the $H_A$ claim that there is lack of fit. Otherwise, fail to reject Ho and state that there is not enough evidence to conclude that there is MLR lack of fit.

Although the lack of fit test seems clever, examining the response plot and residual plot is a much more effective method for examining whether or not the MLR model fits the data well provided that $n \geq 10 p$. A graphical version of the lack of fit test would compute the $\bar{Y}_j$ and see whether they scatter about the identity line in the response plot. When there are no replicates, the range of $\hat{Y}$ could be divided into several narrow nonoverlapping intervals called slices. Then the mean $\bar{Y}_j$ of each slice could be computed and a step function with step height $\bar{Y}_j$ at the $j$ th slice could be plotted. If the step function follows the identity line, then there is no evidence of lack of fit. However, it is easier to check whether the $Y_i$ are scattered about the identity line. Examining the residual plot is useful because it magnifies deviations from the identity line that may be difficult to see until the linear trend is removed. The lack of fit test may be sensitive to the assumption that the errors are iid $N\left(0, \sigma^2\right)$.

# 线性回归分析代写

## 统计代写|线性回归分析代写linear regression analysis代考|Complements

$$\left[\hat{Y}_f+\tilde{r}_L, \hat{Y}_f+\tilde{r}_U\right]$$
Bootstrap PI 需要更多理论而不是使用 $B=1000$ 样 品，使用 $B=\max (1000, n)$. 参见 Olive (2014 年， 第 279-285 页)。

## 统计代写|线性回归分析代写linear regression analysis代考|Lack of Fit Tests

$=S S E-S S P E=\sum_{j=1}^c n_j\left(\bar{Y}j-\hat{Y}_j\right)^2$. 请注意 $\bar{Y}_j$ 是一个无偏估计量 $m\left(\boldsymbol{x}_j\right)$ 尽管 $\hat{Y}_j$ 是一个估计量 $m$ 如 果 MLR 模型合适: $m\left(\boldsymbol{x}_j\right)=\boldsymbol{x}_j^T \boldsymbol{\beta}$. 因此，如果 MLR 模型不合适，SSLF 和 MSLF 可能会非常大。 4 步失拟测试是 i) $\mathrm{Ho}$ ：没有 MLR 失拟的证据， $H_A$ : 不 适合 MLR 模型。 二) $F{L F}=M S L F / M S P E$.
iii) $\mathrm{pval}=P\left(F_{c-p, n-c}>F_{L F}\right)$.
iv) 如果 pval 则拒绝 $\mathrm{Ho} \leq \delta$ 并说明 $H_A$ 声称不合适。否 则，不拒绝 Ho 并声明没有足够的证据得出 MLR 失拟的 结论。

## 有限元方法代写

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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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