# 统计代写|统计推断代写Statistical inference代考|The Delta Method

#### Doug I. Jones

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## 统计代写|统计推断代写Statistical inference代考|The Delta Method

The previous section gives conditions under which a standardized random variable has a limit normal distribution. There are many times, however, when we are not specifically interested in the distribution of the random variable itself, but rather some function of the random variable.

Example 5.5.19 (Estimating the odds) Suppose we observe $X_1, X_2, \ldots, X_n$ independent $\operatorname{Bernoulli}(p)$ random variables. The typical parameter of interest is $p$, the success probability, but another popular parameter is $\frac{p}{1-p}$, the odds. For example, if the data represent the outcomes of a medical treatment with $p=2 / 3$, then a person has odds $2: 1$ of getting better. Moreover, if there were another treatment with success probability $r$, biostatisticians often estimate the odds ratio $\frac{p}{1-p} / \frac{r}{1-r}$, giving the relative odds of one treatment over another.

As we would typically estimate the success probability $p$ with the observed success probability $\hat{p}=\sum_i X_i / n$, we might consider using $\frac{\hat{p}}{1-\hat{p}}$ as an estimate of $\frac{p}{1-p}$. But what are the properties of this estimator? How might we estimate the variance of $\frac{\hat{p}}{1-\hat{p}}$ ? Moreover, how can we approximate its sampling distribution?

Intuition abandons us, and exact calculation is relatively hopeless, so we have to rely on an approximation. The Delta Method will allow us to obtain reasonable, approximate answers to our questions.

One method of proceeding is based on using a Taylor series approximation, which allows us to approximate the mean and variance of a function of a random variable. We will also see that these rather straightforward approximations are good enough to obtain a CLT. We begin with a short review of Taylor series.

## 统计代写|统计推断代写Statistical inference代考|Generating a Random Sample

Thus far we have been concerned with the many methods of describing the behavior of random variables – transformations, distributions, moment calculations, limit theorems. In practice, these random variables are used to describe and model real phenomena, and observations on these random variables are the data that we collect.

Thus, typically, we observe random variables $X_1, \ldots, X_n$ from a distribution $f(x \mid \theta)$ and are most concerned with using properties of $f(x \mid \theta)$ to describe the behavior of the random variables. In this section we are, in effect, going to turn that strategy around. Here we are concerned with generating a random sample $X_1, \ldots, X_n$ from a given distribution $f(x \mid \theta)$.

Example 5.6.1 (Exponential lifetime) Suppose that a particular electrical component is to be modeled with an exponential $(\lambda)$ lifetime. The manufacturer is interested in determining the probability that, out of $c$ components, at least $t$ of them will last $h$ hours. Taking this one step at a time, we have
\begin{aligned} p_1 & =P(\text { component lasts at least } h \text { hours }) \ & =P(X \geq h \mid \lambda), \end{aligned}
and assuming that the components are independent, we can model the outcomes of the $c$ components as Bernoulli trials, so
\begin{aligned} p_2 & =P(\text { at least } t \text { components last } h \text { hours }) \ & =\sum_{k=t}^c\left(\begin{array}{c} c \ k \end{array}\right) p_1^k\left(1-p_1\right)^{c-k} \end{aligned}

# 统计推断代考

## 统计代写|统计推断代写Statistical inference代考|Generating a Random Sample

\begin{aligned} p_1 & =P(\text { component lasts at least } h \text { hours }) \ & =P(X \geq h \mid \lambda), \end{aligned}

\begin{aligned} p_2 & =P(\text { at least } t \text { components last } h \text { hours }) \ & =\sum_{k=t}^c\left(\begin{array}{c} c \ k \end{array}\right) p_1^k\left(1-p_1\right)^{c-k} \end{aligned}

## 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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