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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|回归分析作业代写Regression Analysis代考|Estimation and Practical Use of sigma2

The parameter $\sigma^{2}$ is perhaps the most important parameter of a regression model because it measures prediction accuracy. As shown previously, another way to write the model is $Y=\beta_{0}+\beta_{1} x+\varepsilon$, where $\varepsilon \sim \mathrm{N}\left(0, \sigma^{2}\right)$. Thus the prediction error terms are the $\varepsilon$ values, and these differ from zero with a variance of $\sigma^{2}$.

If the $\beta$ ‘s were known (as is true in simulations but not in reality), you could calculate errors $\varepsilon_{i}=\left{Y_{i}-\left(\beta_{0}+\beta_{1} x_{i}\right)\right}$ and obtain an unbiased estimate of $\sigma^{2}$ as:
(An unbiased estimator of $\left.\sigma^{2}\right)=\frac{1}{n} \sum_{i=1}^{n} \varepsilon_{i}^{2}$
This estimator is unbiased because each individual $\varepsilon_{i}^{2}$ is an unbiased estimator of $\sigma^{2}$, which you can see as follows:
$$\left.\mathrm{E}\left(\varepsilon_{i}^{2}\right)=\mathrm{E} \mid\left(Y_{i}-\beta_{0}-\beta_{1} x_{i}\right)^{2}\right)=\operatorname{Var}\left(Y_{i} \mid X=x_{i}\right)=\sigma^{2} .$$

However, in practice, you cannot use this estimator because the $\beta$ ‘s are unknown; thus the $\varepsilon^{\prime}$ s are unknown (or unobservable) as well. But you can use a similar estimator based on the residuals $e_{i}=\left{Y_{i}-\left(\hat{\beta}{0}+\hat{\beta}{1} x_{i}\right)\right}$, which are observable:
(Another estimator of $\sigma^{2}$ ) $=\frac{1}{n} \sum_{i=1}^{n} e_{i}^{2}$
This is, in fact, the maximum likelihood estimator, as given in Chapter 2. However, this estimator is biased: Recall that the values $\hat{\beta}{0}, \hat{\beta}{1}$ are chosen to minimize SSE; that is, $\mathrm{SSE}=\sum_{i=1}^{n} e_{i}^{2}$ is a minimum. In particular, $\mathrm{SSE}=\sum_{i=1}^{n} e_{i}^{2} \leq \sum_{i=1}^{n} \varepsilon_{i}^{2}$, which means that the estimator $\frac{1}{n} \sum_{i=1}^{n} e_{i}^{2}=\mathrm{SSE} / n$ is biased low since $\frac{1}{n} \sum_{i=1}^{n} \varepsilon_{i}^{2}$ is unbiased.

In basic statistics, you learned that the variance estimator uses “n-1″ in the denominator instead of ” $n$ ” to remove similar bias; the quantity ” $n-1$ ” is sometimes called degrees of freedom. You may have also heard that you lose a degree of freedom for every parameter you estimate. In regression, these parameters refer to the $\beta^{\prime}$ ‘s, so in simple regression, you lose two degrees of freedom. This leads to the following estimator of $\sigma^{2}$.

## 统计代写|回归分析作业代写Regression Analysis代考|Standard Errors

The Gauss-Markov theorem states that the OLS estimator has minimum variance among linear unbiased estimators. What does “variance” of the OLS estimator refer to? Please look at Figure $3.1$ again: You can see that there is variability in the possible values of $\hat{\beta}{1}$ ranging from $1.0$ to $2.0$. Variance of the estimator $\hat{\beta}{1}$, denoted symbolically by $\operatorname{Var}\left(\hat{\beta}{1}\right)$, refers to the variance of the distribution $p\left(\hat{\beta}{1}\right)$ that is shown in Figure 3.1.

If the assumptions of the Gauss-Markov model are true, then the following formula gives the exact variance of the OLS estimator $\hat{\beta}{1}$. Variance of the OLS estimator $\hat{\beta}{1}$
$$\operatorname{Var}\left(\hat{\beta}{1}\right)=\frac{\sigma^{2}}{(n-1) s{x}^{2}}$$
In the formula for $\operatorname{Var}\left(\beta_{1}\right)$, note that $s_{x}^{2}=\sum\left(x_{i}-\hat{\mu}{x}\right)^{2} /(n-1)$ is the usual estimate of the variance of $X$. Note that the $\operatorname{Var}\left(\hat{\beta}{1}\right)$ formula is conditional on the observed values of the $X$ data; this is apparent because $s_{x}^{2}$ is specifically a function of the observed $X$ data.

When coupled with unbiasedness of $\hat{\beta}{1}$, smaller $\operatorname{Var}\left(\hat{\beta}{1}\right)$ implies a more accurate estimate, i.e., an estimate that tends to be closer to $\beta_{1}$. Hence, we have the following interesting conclusions regarding the accuracy of the OLS estimate $\hat{\beta}{1}$ : The OLS estimate $\hat{\beta}{1}$ of $\beta_{1}$ is more accurate when:

• $n$ is larger, and/or
• $\mathrm{s}_{x}^{2}$ is larger, and/or
• $\sigma^{2}$ is smaller.
As mentioned above, the formula given for $\operatorname{Var}\left(\hat{\beta}{1}\right)$ can be mathematically derived from the assumptions of the Gauss-Markov model. Violation of assumptions renders the formula incorrect. In particular, violation of the homoscedasticity assumption is the rationale for using heteroscedasticity-consistent standard errors, which are covered in Chapter $12 .$ Strangely enough, the mathematics needed to prove the variance formula is easier in the multiple regression model, so we will prove it later in Chapter 7. But for now, you should understand the assumptions that imply the result (e.g., the classical model) and the result itself (the formula for $\operatorname{Var}\left(\hat{\beta}{1}\right)$ ) by using simulation: If you simulate many thousands of data sets from the same model, with the same sample size, and with the same $X$ data, then the sample variance estimate of the resulting thousands of $\hat{\beta}{1}$ estimates will be (within simulation error) equal to $\sigma^{2} /\left{(n-1) s{x}^{2}\right}$. The simulation also clarifies the “conditional on observed values of the $X$ data” interpretation because the $X$ data are the same for every simulated data set.

# 回归分析代写

## 统计代写|回归分析作业代写Regression Analysis代考|Estimation and Practical Use of sigma2

$$\left.\mathrm{E}\left(\varepsilon_{i}^{2}\right)=\mathrm{E} \mid\left(Y_{i}-\beta_{0}-\beta_{1} x_{i}\right)^{2}\right)=\operatorname{Var}\left(Y_{i} \mid X=x_{i}\right)=\sigma^{2} .$$

(left 的分隔符缺失或无法识别 另一个估计 $\left.\sigma^{2}\right)=\frac{1}{n} \sum_{i=1}^{n} e_{i}^{2}$

## 统计代写|回归分析作业代写Regression Analysis代考|Standard Errors

$$\operatorname{Var}(\hat{\beta} 1)=\frac{\sigma^{2}}{(n-1) s x^{2}}$$ 到的函数 $X$ 数据。

• $n$ 更大, 和/或
• $s_{x}^{2}$ 更大，和/或
• $\sigma^{2}$ 更小。
如上所述，给出的公式为 $\operatorname{Var}(\hat{\beta} 1)$ 可以从高斯-马尔可夫模型的假设数学推导出来。违反假设会使公式不正确。特别是，违反同方差假设是使用异方差一致 标准误的基本原理，这将在第 1 章中介绍。12.奇怪的是，证明方差公式所需的数学在多元回归模型中更容易，因此我们将在第 7 章后面证明它。但是现 在，您应该了解暗示结果的假设 (例如，经典模型) 和结果本身 (公式为 $\operatorname{Var}(\hat{\beta} 1)$ ) 通过使用模拟：如果您从相同的模型、相同的样本量和相同的 $X$ 数
据，然后对结果的数干个样本方差估计 $\hat{\beta} 1$ 估计将（在模拟误差内) 等于 $\backslash 1 \mathrm{eft}$ 的分隔符缺失或无法识别
模拟还阐明了“以观测值为条件的 $X$ 数据”的解释，因为 $X$ 每个模拟数据集的数据都是相同的。

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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
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