### 金融代写|量化风险管理代写Quantitative Risk Management代考|PROJMGNT 5004

statistics-lab™ 为您的留学生涯保驾护航 在代写量化风险管理Quantitative Risk Management方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写量化风险管理Quantitative Risk Management代写方面经验极为丰富，各种代写量化风险管理Quantitative Risk Management相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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

## 金融代写|量化风险管理代写Quantitative Risk Management代考|Distance Between Representative Values

Three main measures are constituting this group:

• In statistics, the range is simply the difference between the highest and lowest value taken by the variable under consideration, but it might have a more complex meaning (see below).
1. For $n$ independent and identically distributed continuous random variables $X_{1}, X_{2}, \ldots, X_{n}$ with cumulative distribution function $F(x)$ and probability density function $f(x)$, let $t$ denote the range of a sample of size $n$ from a population with distribution function $F(x)$.
The range has cumulative distribution function (Gumbel 1947)
$$G(t)=n \int_{-\infty}^{\infty} f(x)[F(x+t)-F(x)]^{n-1} \mathrm{~d} x$$
The mean range is given as follows (Hartley and David 1954):
$$n \int_{0}^{1} F^{-1}\left[F^{n-1}-(1-F)^{n-1}\right] \mathrm{d} F$$
2. For $n$ non-identically distributed independent continuous random variables $X_{1}, X_{2}, \ldots, X_{n}$ with cumulative distribution functions $F_{1}(x), F_{2}(x), \ldots$, $F_{n}(x)$ and probability density functions $f_{1}(x), f_{2}(x), \ldots, f_{n}(x)$, the range has cumulative distribution function (Tsimashenka et al. 2012)
$$G(t)=\sum_{i=1}^{n} \int_{-\infty}^{\infty} f_{i}(x) \prod_{j=1, j \neq i}^{n}\left[F_{j}(x+t)-F_{j}(x)\right] \mathrm{d} x .$$
3. For $n$ independent and identically distributed discrete random variables $X_{1}, X_{2}, \ldots, X_{n}$ with cumulative distribution function $F(x)$ and probability mass function $f(x)$ the range of the $X_{i}$ is the range of a sample of size $n$ from a population with distribution function $F(x)$.

The range has probability mass function as follows (Evans et al. 2006; Burr 1955; Abdel-Aty 1954; Siotani 1956):
$$g(t)=\left{\begin{array}{l} \sum_{x=1}^{N}[f(x)]^{n} \ \sum_{x=1}^{N-1}\left(\begin{array}{l} {[F(x+t)-F(x-1)]^{n}} \ -[F(x+t)-F(x)]^{n} \ -[F(x+t-1)-F(x-1)]^{n} \ +[F(x+t-1)-F(x)]^{n} \end{array}\right) \quad t=0 \end{array} \quad t=1,2,3 \ldots, N-1 .\right.$$

## 金融代写|量化风险管理代写Quantitative Risk Management代考|The Variance

The variance and its square root, i.e., the standard deviation, constitute the most widely employed measures. The variance is defined as the expected value of the squared deviations of the data values from the mean, and thus simply measures the dispersion of the estimates around their mean value. Let $X$ be a random variable defined on the probability space previously introduced, then the expected value of

$X$, denoted by $E[X]$, is defined as the Lebesgue integral
$$E[X]=\int_{\Omega} X(\omega) d \mathrm{P}(\omega) .$$
In our case, the expected value corresponds to the mean. Formally, the variance of a random variable $X$ is the expected value of the squared deviation from the mean of $\mu=\mathrm{E}[X]$
$$\operatorname{Var}(X)=\mathrm{E}\left[(X-\mu)^{2}\right]$$
The variance is also the second moment or second cumulant of a probability distribution that generates $X$. The variance is typically designated as $\operatorname{Var}(X), \sigma_{X}^{2}$, $\sigma^{2}$. The expression for the variance can be expanded as follows:
\begin{aligned} \operatorname{Var}(X) &=\mathrm{E}\left[(X-\mathrm{E}[X])^{2}\right] \ &=\mathrm{E}\left[X^{2}-2 X \mathrm{E}[X]+\mathrm{E}[X]^{2}\right] \ &=\mathrm{E}\left[X^{2}\right]-2 \mathrm{E}[X] \mathrm{E}[X]+\mathrm{E}[X]^{2} \ &=\mathrm{E}\left[X^{2}\right]-\mathrm{E}[X]^{2} \end{aligned}
If the random variable $X$ follows a continuous distribution with probability density function $f(x)$, then the variance of $X$ is given by
\begin{aligned} \operatorname{Var}(X) &=\sigma^{2} \ &=\int(x-\mu)^{2} f(x) d x \ &=\int x^{2} f(x) d x-2 \mu \int x f(x) d x+\int \mu^{2} f(x) d x \ &=\int x^{2} f(x) d x-\mu^{2} \end{aligned}
where $\mu$ is the expected value of $X$ given by the following:
$$\mu=\int x f(x) d x,$$
and where $x$ is ranging over the range of $X$.

## 金融代写|量化风险管理代写Quantitative Risk Management代考|The Expected Absolute Deviation

The expected absolute deviation (sometimes called the mean absolute deviation) is the sum of the absolute values of the deviations from the mean (of course this measure could be adapted to any other threshold, like 0 , the median or the mode, for example ${ }^{1}$ ).

The term average absolute deviation does not uniquely identify a measure of statistical dispersion, as there are several measures that can be used to measure absolute deviations, and there are several measures of central tendency that can be used as well. Thus, to uniquely identify the absolute deviation it is necessary to specify both the measure of deviation and the measure of central tendency. Unfortunately, the statistical literature has not yet adopted a standard notation, as both the mean absolute deviation around the mean and the median absolute deviation around the median have been denoted by their initials “MAD” in the literature, which may lead to confusion, since in general, they may have values considerably different from each other.

The mean absolute deviation of a set $x_{1}, x_{2}, \ldots, x_{n}$ issued of a r.v. X, is given by the following equation:
$$\mathbb{E}(|X-m(X)|)=\frac{1}{n} \sum_{i=1}^{n}\left|x_{i}-m(X)\right|$$
where $m(X)$ represent the chosen central tendency, usually the median, the mode, or the mean of the r.v. X. It is noteworthy to mention that the choice of the central tendency impacts the metric.

The mean absolute deviation from the median is less than or equal to the mean absolute deviation from the mean. In fact, the mean absolute deviation from the median is always less than or equal to the mean absolute deviation from any other fixed number. The mean absolute deviation from the mean (denoted $\mu$ in what follows) is less than or equal to the standard deviation; one way of proving this relies on Jensen’s inequality: $\phi(\mathbb{E}[Y]) \leq \mathbb{E}[\phi(Y)]$, where $\phi$ is a convex function, this implies for $Y=|X-\mu| \mu$ being the sample mean that:
$$\begin{gathered} \mathbb{E}(|X-\mu|)^{2} \leq \mathbb{E}\left(|X-\mu|^{2}\right) \ \mathbb{E}(|X-\mu|)^{2} \leq \operatorname{Var}(X) \end{gathered}$$

## 金融代写|量化风险管理代写Quantitative Risk Management代考|Distance Between Representative Values

• 在统计学中，范围只是所考虑变量的最高值和最低值之间的差，但它可能具有更复杂的含义（见下文）。
1. 为了n独立同分布的连续随机变量X1,X2,…,Xn具有累积分布函数F(X)和概率密度函数F(X)， 让吨表示大小样本的范围n来自具有分布函数的总体F(X).
范围具有累积分布函数 (Gumbel 1947)
G(吨)=n∫−∞∞F(X)[F(X+吨)−F(X)]n−1 dX
平均范围如下（Hartley and David 1954）：
n∫01F−1[Fn−1−(1−F)n−1]dF
2. 为了n非同分布独立连续随机变量X1,X2,…,Xn具有累积分布函数F1(X),F2(X),…, Fn(X)和概率密度函数F1(X),F2(X),…,Fn(X), 范围具有累积分布函数 (Tsimashenka et al. 2012)
G(吨)=∑一世=1n∫−∞∞F一世(X)∏j=1,j≠一世n[Fj(X+吨)−Fj(X)]dX.
3. 为了n独立同分布的离散随机变量X1,X2,…,Xn具有累积分布函数F(X)和概率质量函数F(X)的范围X一世是大小样本的范围n来自具有分布函数的总体F(X).

$$g(t)=\left{ \begin{array}{l} \sum_{x=1}^{N}[f(x)]^{n} \ \sum_{x=1}^{N-1}\left(\begin{array }{l} {[F(x+t)-F(x-1)]^{n}} \ -[F(x+t)-F(x)]^{n} \ -[F(x +t-1)-F(x-1)]^{n} \ +[F(x+t-1)-F(x)]^{n} \end{数组}\begin{array}{l} \sum_{x=1}^{N}[f(x)]^{n} \ \sum_{x=1}^{N-1}\left(\begin{array }{l} {[F(x+t)-F(x-1)]^{n}} \ -[F(x+t)-F(x)]^{n} \ -[F(x +t-1)-F(x-1)]^{n} \ +[F(x+t-1)-F(x)]^{n} \end{数组}\right) \quad t=0 \end{array} \quad t=1,2,3 \ldots, N-1 .\right.$$

## 金融代写|量化风险管理代写Quantitative Risk Management代考|The Variance

X，表示为和[X], 被定义为勒贝格积分

μ=∫XF(X)dX,

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。