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# 数据科学代写|假设检验代考Hypothesis Testing代考|STAT1070 Basic Tools for Judging Robustness

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## 数据科学代写|假设检验代考Hypothesis Testing代考|Basic Tools for Judging Robustness

There are three basic tools used to establish whether quantities such as measures of location and scale have good properties: qualitative robustness, quantitative robustness, and infinitesimal robustness. This section describes these tools in the context of location measures, but they are relevant to measures of scale, as will become evident. These tools not only provide formal methods for judging a particular measure, they can be used to help derive measures that are robust.

Before continuing, it helps to be more formal about what is meant by a measure of location. A quantity that characterizes a distribution, such as the population mean, is said to be a measure of location if it satisfies four conditions, and a fifth is sometimes added. To describe them, let $X$ be a random variable with distribution $F$, and let $\theta(X)$ be some descriptive measure of $F$. Then $\theta(X)$ is said to be a measure of location if for any constants $a$ and $b$,

The first condition is called location equivariance. It simply requires that if a constant $b$ is added to every possible value of $X$, a measure of location should be increased by the same amount. Let $E(X)$ denote the expected value of $X$. From basic principles, the population mean is location equivariant. That is, if $\theta(X)=E(X)=\mu$, then $\theta(X+b)=E(X+b)=\mu+b$. The first three conditions, taken together, imply that a measure of location should have a value within the range of possible values of $X$. The fourth condition is called scale equivariance. If the scale by which something is measured is altered by multiplying all possible values of $X$ by $a$, a measure of location should be altered by the same amount. In essence, results should be independent of the scale of measurement. As a simple example, if the typical height of a man is to be compared to the typical height of a woman, it should not matter whether the comparisons are made in inches or feet.

## 数据科学代写|假设检验代考Hypothesis Testing代考|Qualitative Robustness

To understand qualitative robustness, it helps to begin by considering any function $f(x)$, not necessarily a probability density function. Suppose it is desired to impose a restriction on this function so that it does not change drastically with small changes in $x$. One way of doing this is to insist that $f(x)$ be continuous. If, for example, $f(x)=0$ for $x \leq 1$, but $f(x)=10,000$ for any $x>1$, the function is not continuous, and if $x=1$, an arbitrarily small increase in $x$ results in a large increase in $f(x)$.

A similar idea can be used when judging a measure of location. This is accomplished by viewing parameters as functionals. In the present context, a functional is just a rule that maps every distribution into a real number. For example, the population mean can be written as
$$T(F)=E(X),$$
where the expected value of $X$ depends on $F$. The role of $F$ becomes more explicit if expectation is written in integral form, in which case this last equation becomes
$$T(F)=\int_{-\infty}^{\infty} x d F(x) .$$
If $X$ is discrete and the probability function corresponding to $F(x)$ is $f(x)$,
$$T(F)=\sum x f(x),$$
where the summation is over all possible values $x$ of $X$.
One advantage of viewing parameters as functionals is that the notion of continuity can be extended to them. Thus, if the goal is to have measures of location that are relatively unaffected by small shifts in $F$, a requirement that can be imposed is that when viewed as a functional, it is continuous. Parameters with this property are said to have qualitative robustness.

## 数据科学代写|假设检验代考Hypothesis Testing代考|Qualitative Robustness

$$T(F)=E(X),$$

$$T(F)=\int_{-\infty}^{\infty} x d F(x) .$$

$$T(F)=\sum x f(x),$$

## 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。