### 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|STAT302

statistics-lab™ 为您的留学生涯保驾护航 在代写多元统计分析Multivariate Statistical Analysis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。我们的专家在代写多元统计分析Multivariate Statistical Analysis代写方面经验极为丰富，各种代写多元统计分析Multivariate Statistical Analysis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

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Multivariate Laplace Distribution

Let $g$ and $G$ be the pdf and cdf of a $d$-dimensional Gaussian distribution $N_{d}(0, \Sigma)$, the pdf and cdf of a multivariate Laplace distribution can be written as
$f_{M \text { Laplace } d}(x ; m, \Sigma)=\int_{0}^{\infty} g\left(z^{-\frac{1}{2}} x-z^{\frac{1}{2}} m\right) z^{-\frac{d}{2}} e^{-z} d z$
$F_{M \text { Laplace }{d}}(x, m, \Sigma)=\int{0}^{\infty} G\left(z^{-\frac{1}{2}} x-z^{\frac{1}{2}} m\right) e^{-z} d z$
the pdf can also be described as
\begin{aligned} f_{M \text { Laplace }{d}}(x ; m, \Sigma)=& \frac{2 e^{x^{\top} \Sigma^{-1} m}}{(2 \pi)^{\frac{d}{2}}|\Sigma|^{\frac{1}{2}}}\left(\frac{x^{\top} \Sigma^{-1} x}{2+m^{\top} \Sigma^{-1} m}\right)^{\frac{2}{2}} \ & \times K{\lambda}\left(\sqrt{\left(2+m^{\top} \Sigma^{-1} m\right)\left(x^{\top} \Sigma^{-1} x\right)}\right) \end{aligned}
where $\lambda=\frac{2-d}{2}$ and $K_{\lambda}(x)$ is the modified Bessel function of the third kind
$$K_{\lambda}(x)=\frac{1}{2}\left(\frac{x}{2}\right)^{\lambda} \int_{0}^{\infty} t^{-\lambda-1} e^{-t-\frac{t^{2}}{4 t}} d t, \quad x>0$$
Multivariate Laplace distribution has mean and variance
\begin{aligned} \mathrm{E}[X] &=m \ \operatorname{Cov}[X] &=\Sigma+m m^{\top} \end{aligned}

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Copulae

The cumulative distribution function (cdf) of a two-dimensional vector $\left(X_{1}, X_{2}\right)$ is given by
$$F\left(x_{1}, x_{2}\right)=\mathrm{P}\left(X_{1} \leq x_{1}, X_{2} \leq x_{2}\right)$$
For the case that $X_{1}$ and $X_{2}$ are independent, their joint cumulative distribution function $F\left(x_{1}, x_{2}\right)$ can be written as a product of their one-dimensional marginals:
$$F\left(x_{1}, x_{2}\right)=F_{X_{1}}\left(x_{1}\right) F_{X_{2}}\left(x_{2}\right)=\mathrm{P}\left(X_{1} \leq x_{1}\right) \mathrm{P}\left(X_{2} \leq x_{2}\right) .$$
But how can we model dependence of $X_{1}$ and $X_{2}$ ? Most people would suggest linear correlation. Correlation is though an appropriate measure of dependence only when the random variables have an elliptical or spherical distribution, which include the normal multivariate distribution. Although the terms “correlation” and “dependency” are often used interchangeably, correlation is actually a rather imperfect measure of dependency, and there are many circumstances where correlation should not be used.

Copulae represent an elegant concept of connecting marginals with joint cumulative distribution functions. Copulae are functions that join or “couple” multivariate distribution functions to their one-dimensional marginal distribution functions. Let us consider a $d$-dimensional vector $X=\left(X_{1}, \ldots, X_{d}\right)^{\top}$. Using copulae, the marginal distribution functions $F_{X_{i}}(i=1, \ldots, d)$ can be separately modelled from their dependence structure and then coupled together to form the multivariate distribution $F_{X}$. Copula functions have a long history in probability theory and statistics. Their application in finance is very recent. Copulae are important in Valueat-Risk calculations and constitute an essential tool in quantitative finance (Härdle et al., 2009).

First let us concentrate on the two-dimensional case, then we will extend this concept to the $d$-dimensional case, for a random variable in $\mathbb{R}^{d}$ with $d \geq 1$. To be able to define a copula function, first we need to represent a concept of the volume of a rectangle, a 2 -increading function and a grounded function.

Let $U_{1}$ and $U_{2}$ be two sets in $\overline{\mathbb{R}}=\mathbb{R} \cup{+\infty} \cup{-\infty}$ and consider the function $F: U_{1} \times U_{2} \longrightarrow \mathbb{\mathbb { R }}$.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Elementary Properties of the Multinormal

Let us first summarise some properties which were already derived in the previous chapter.

• The pdf of $X \sim N_{p}(\mu, \Sigma)$ is
$$f(x)=|2 \pi \Sigma|^{-1 / 2} \exp \left{-\frac{1}{2}(x-\mu)^{\top} \Sigma^{-1}(x-\mu)\right}$$

The expectation is $\mathrm{E}(X)=\mu$, the covariance can be calculated as $\operatorname{Var}(X)=\mathrm{E}(X-\mu)(X-\mu)^{\top}=\Sigma$.

Linear transformations turn normal random variables into normal random variables. If $X \sim N_{p}(\mu, \Sigma)$ and $\mathcal{A}(p \times p), c \in \mathbb{R}^{p}$, then $Y=\mathcal{A} X+c$ is $p$-variate Normal, i.e.
$$Y \sim N_{p}\left(\mathcal{A} \mu+c, \mathcal{A} \Sigma \cdot \mathcal{A}^{\top}\right)$$

If $X \sim N_{p}(\mu, \Sigma)$, then the Mahalanobis transformation is
$$Y=\Sigma^{-1 / 2}(X-\mu) \sim N_{p}\left(0, \mathcal{I}{p}\right)$$ and it holds that $$Y^{\top} Y=(X-\mu)^{\top} \Sigma^{-1}(X-\mu) \sim \chi{p}^{2}$$
Often it is interesting to partition $X$ into sub-vectors $X_{1}$ and $X_{2}$. The following theorem tells us how to correct $X_{2}$ to obtain a vector which is independent of $X_{1}$.
Theorem 5.1 Let $X=\left(\begin{array}{l}X_{1} \ X_{2}\end{array}\right) \sim N_{p}(\mu, \Sigma), X_{1} \in \mathbb{R}^{r}, X_{2} \in \mathbb{R}^{p-r}$. Define $X_{2.1}=$ $X_{2}-\Sigma_{21} \Sigma_{11}^{-1} X_{1}$ from the partitioned covariance matrix
$$\Sigma=\left(\begin{array}{cc} \Sigma_{11} & \Sigma_{12} \ \Sigma_{21} & \Sigma_{22} \end{array}\right)$$
Then
$$\begin{array}{r} X_{1} \sim N_{r}\left(\mu_{1}, \Sigma_{11}\right), \ X_{2.1} \sim N_{p-r}\left(\mu_{2.1}, \Sigma_{22.1}\right) \end{array}$$
are independent with
$$\mu_{2.1}=\mu_{2}-\Sigma_{21} \Sigma_{11}^{-1} \mu_{1}, \quad \Sigma_{22.1}=\Sigma_{22}-\Sigma_{21} \Sigma_{11}^{-1} \Sigma_{12}$$
Proof
$$\begin{array}{rlll} X_{1} & =\mathcal{A} X & \text { with } & \mathcal{A}=\left(\mathcal{I}{r}, 0\right) \ X{2.1} & =\mathcal{B} X & \text { with } & \mathcal{B}=\left(-\Sigma_{21} \Sigma_{11}^{-1}, \mathcal{I}_{p-r}\right) \end{array}$$

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Multivariate Laplace Distribution

F米 拉普拉斯 d(X;米,Σ)=∫0∞G(和−12X−和12米)和−d2和−和d和
F米 拉普拉斯 d(X,米,Σ)=∫0∞G(和−12X−和12米)和−和d和
pdf也可以描述为

F米 拉普拉斯 d(X;米,Σ)=2和X⊤Σ−1米(2圆周率)d2|Σ|12(X⊤Σ−1X2+米⊤Σ−1米)22 ×ķλ((2+米⊤Σ−1米)(X⊤Σ−1X))

ķλ(X)=12(X2)λ∫0∞吨−λ−1和−吨−吨24吨d吨,X>0

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Copulae

F(X1,X2)=磷(X1≤X1,X2≤X2)

F(X1,X2)=FX1(X1)FX2(X2)=磷(X1≤X1)磷(X2≤X2).

Copulae 代表了一个优雅的概念，它将边缘与联合累积分布函数连接起来。Copulae 是将多元分布函数连接或“耦合”到其一维边际分布函数的函数。让我们考虑一个d维向量X=(X1,…,Xd)⊤. 使用 copulae，边际分布函数FX一世(一世=1,…,d)可以从它们的依赖结构中单独建模，然后耦合在一起形成多元分布FX. Copula 函数在概率论和统计学中有着悠久的历史。它们在金融领域的应用是最近才出现的。Copulae 在 Valueat-Risk 计算中很重要，并且构成了量化金融中的重要工具（Härdle 等，2009）。

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Elementary Properties of the Multinormal

• 的pdfX∼ñp(μ,Σ)是
f(x)=|2 \pi \Sigma|^{-1 / 2} \exp \left{-\frac{1}{2}(x-\mu)^{\top} \Sigma^{-1 }(x-\mu)\right}f(x)=|2 \pi \Sigma|^{-1 / 2} \exp \left{-\frac{1}{2}(x-\mu)^{\top} \Sigma^{-1 }(x-\mu)\right}

Σ=(Σ11Σ12 Σ21Σ22)

X1∼ñr(μ1,Σ11), X2.1∼ñp−r(μ2.1,Σ22.1)

μ2.1=μ2−Σ21Σ11−1μ1,Σ22.1=Σ22−Σ21Σ11−1Σ12

X1=一个X 和 一个=(我r,0) X2.1=乙X 和 乙=(−Σ21Σ11−1,我p−r)

## 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。