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# 统计代写|假设检验代考Hypothesis Testing代考|STAT311 A Bootstrap Estimate of a Standard Error

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## 统计代写|假设检验代考Hypothesis Testing代考|A Bootstrap Estimate of a Standard Error

It is convenient to begin with a description of the most basic bootstrap method for estimating a standard error. Let $\hat{\theta}$ be any estimator based on a random sample of observations, $X_1, \ldots, X_n$. The goal is to estimate $\operatorname{VAR}(\hat{\theta})$, the squared standard error of $\hat{\theta}$. The strategy used by the bootstrap method is based on a very simple idea. Temporarily assume that observations are randomly sampled from some known distribution, $F$. Then for a given sample size, $n$, the sampling distribution of $\hat{\theta}$ could be determined by randomly generating $n$ observations from $F$, computing $\hat{\theta}$, randomly generating another set of $n$ observations, computing $\hat{\theta}$, and repeating this process many times. Suppose this is done $B$ times and the resulting values for $\hat{\theta}$ are labeled $\hat{\theta}1, \ldots, \hat{\theta}_B$. If $B$ is large enough, the values $\hat{\theta}_1, \ldots, \hat{\theta}_B$ provide a good approximation of the distribution of $\hat{\theta}$. In particular, they provide an estimate of the squared standard error of $\hat{\theta}$, namely, $$\frac{1}{B-1} \sum{b=1}^B\left(\hat{\theta}b-\bar{\theta}\right)^2,$$ where $$\bar{\theta}=\frac{1}{B} \sum{b=1}^B \hat{\theta}_b .$$
That is, $\operatorname{VAR}(\hat{\theta})$ is estimated with the sample variance of the values $\hat{\theta}_1, \ldots, \hat{\theta}_B$. If, for example, $\hat{\theta}$ is taken to be the sample mean, $\bar{X}$, then the squared standard error would be found to be $\sigma^2 / n$, approximately, provided $B$ is reasonably large. Of course when working with the mean, it is known that its squared standard error is $\sigma^2 / n$, so the method just described is unnecessary. The only point is that a reasonable method for estimating the squared standard error of $\hat{\theta}$ has been described.

## 统计代写|假设检验代考Hypothesis Testing代考|R and S-PLUS Function bootse

As explained in Section 1.7, $\mathrm{R}$ and S-PLUS functions have been written for applying the methods described in this book. The software written for this book is free, and a single command incorporates them into your version of $\mathrm{R}$ or S-PLUS. Included is the function
$$\text { bootse ( } x, n \text { boot }=1000 \text {, est=median }) \text {, }$$
which can be used to compute a bootstrap estimate of the standard error of virtually any estimator covered in this book. Here, $\mathrm{x}$ is any $\mathrm{R}$ (or S-PLUS) variable containing the data. The argument nboot represents $B$, the number of bootstrap samples, and defaults to 1000 if not specified. (As is done with all $\mathrm{R}$ and S-PLUS functions, optional arguments are indicated by $a \mathrm{n}=$ and they default to the value shown. Here, for example, the value of nboot is taken to be 1000 if no value is specified by the user.) The argument est indicates the estimator for which the standard error is to be computed. If not specified, est defaults to the median. That is, the standard error of the usual sample median will be estimated. So, for example, if data are stored in the R variable blob, the command bootse (blob) will return the estimated standard error of the usual sample median.

## 统计代写|假设检验代考Hypothesis Testing代考|R and S-PLUS Function bootse

$$\text { bootse }(x, n \text { boot }=1000 \text {, est=median }) \text {, }$$

## 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。