Posted on Categories:Hypothesis Testing, 假设检验, 数据科学代写, 统计代写, 统计代考

# 数据科学代写|假设检验代考Hypothesis Testing代考|STA2023 The Influence Curve

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数据科学代写|假设检验代考Hypothesis Testing代考|The Influence Curve

This section gives one more indication of why robust methods are of interest by introducing the influence curve, as described by Mosteller and Tukey (1977). It bears a close resemblance to the influence function, which plays an important role in subsequent chapters, but the influence curve is easier to understand. In general, the influence curve indicates how any statistic is affected by an additional observation having value $x$. In particular it graphs the value of a statistic versus $x$.

As an illustration, let $\bar{X}$ be the sample mean corresponding to the random sample $X_1, \ldots, X_n$. Suppose we add an additional value, $x$, to the $n$ values already available, so now there are $n+1$ observations. Of course this additional value will in general affect the sample mean, which is now $\left(x+\sum X_i\right) /(n+1)$. It is evident that as $x$ gets large, the sample mean of all $n+1$ observations increases. The influence curve plots $x$ versus
$$\frac{1}{n+1}\left(x+\sum X_i\right)$$
the idea being to illustrate how a single value can influence the value of the sample mean. Note that for the sample mean, the graph is a straight line with slope $1 /(n+1)$, the point being that the curve increases without bound. Of course, as $n$ gets large, the slope decreases, but in practice there might be two or more unusual values that dominate the value of $\bar{X}$.

Now consider the usual sample median, $M$. Let $X_{(1)} \leq \cdots \leq X_{(n)}$ be the observations written in ascending order. If $n$ is odd, let $m=(n+1) / 2$, in which case $M=X_{(m)}$, the $m$ th largest-order statistic. If $n$ is even, let $m=n / 2$, in which case $M=\left(X_{(m)}+X_{(m+1)}\right) / 2$. To be more concrete, consider the values

$\begin{array}{llllllllll}2 & 4 & 6 & 7 & 8 & 10 & 14 & 19 & 2128 .\end{array}$
Then $n=10$ and $M=(8+10) / 2=9$. Suppose an additional value, $x$, is added so that now $n=11$. If $x>10$, then $M=10$, regardless of how large $x$ might be. If $x<8, M=8$ regardless of how small $x$ might be. As $x$ increases from 8 to $10, M$ increases from 8 to 10 as well. The main point is that in contrast to the sample mean, the median has a bounded influence curve. In general, if the goal is to minimize the influence of a relatively small number of observations on a measure of location, attention might be restricted to those measures having a bounded influence curve. A concern with the median, however, is that its standard error is large relative to the standard error of the mean when sampling from a normal distribution, so there is interest in searching for other measures of location having a bounded influence curve but that have reasonably small standard errors when distributions are normal.

## 数据科学代写|假设检验代考Hypothesis Testing代考|The Central Limit Theorem

When working with means or least squares regression, certainly the bestknown method for dealing with nonnormality is to appeal to the central limit theorem. Put simply, under random sampling, if the sample size is sufficiently large, the distribution of the sample mean is approximately normal under fairly weak assumptions. A practical concern is the description sufficiently large. Just how large must $n$ be to justify the assumption that $\bar{X}$ has a normal distribution? Early studies suggested that $n=40$ is more than sufficient, and there was a time when even $n=25$ seemed to suffice. These claims were not based on wild speculations, but more recent studies have found that these early investigations overlooked two crucial aspects of the problem.

The first is that early studies looking into how quickly the sampling distribution of $\bar{X}$ approaches a normal distribution focused on very light-tailed distributions, where the expected proportion of outliers is relatively low. In particular, a popular way of illustrating the central limit theorem was to consider the distribution of $\bar{X}$ when sampling from a uniform or exponential distribution. These distributions look nothing like a normal curve, the distribution of $\bar{X}$ based on $n=40$ is approximately normal, so a natural speculation is that this will continue to be the case when sampling from other nonnormal distributions. But more recently it has become clear that as we move toward more heavy-tailed distributions, a larger sample size is required.

The second aspect being overlooked is that when making inferences based on Student’s $t$, the distribution of $t$ can be influenced more by nonnormality than the distribution of $\bar{X}$. Even when sampling from a relatively light-tailed distribution, practical problems arise when using Student’s $\mathrm{t}$, as will be illustrated in Section 4.1. When sampling from heavy-tailed distributions, even $n=300$ might not suffice when computing a $.95$ confidence interval.

## 数据科学代写|假设检验代考Hypothesis Testing代考|The Influence Curve

$$\frac{1}{n+1}\left(x+\sum X_i\right)$$

$\begin{array}{llllllllll}2 & 4 & 6 & 7 & 8 & 10 & 14 & 19 & 2128 .\end{array}$

## 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。