### 统计代写|应用时间序列分析代写applied time series analysis代考|Scale Mixture of Normal Distributions

statistics-lab™ 为您的留学生涯保驾护航 在代写应用时间序列分析applied time series 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。我们的专家在代写应用时间序列分析applied time series analysis方面经验极为丰富，各种代写应用时间序列分析applied time series analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|应用时间序列分析代写applied time series anakysis代考|Scale Mixture of Normal Distributions

Recent studies of stock returns tend to use scale mixture or finite mixture of normal distributions. Under the assumption of scale mixture of normal distributions, the log return $r_{t}$ is normally distributed with mean $\mu$ and variance $\sigma^{2}\left[\right.$ i.e., $\left.r_{t} \sim N\left(\mu, \sigma^{2}\right)\right]$. However, $\sigma^{2}$ is a random variable that follows a positive distribution (e.g., $\sigma^{-2}$ follows a Gamma distribution). An example of finite mixture of normal distributions is
$$r_{t} \sim(1-X) N\left(\mu, \sigma_{1}^{2}\right)+X N\left(\mu, \sigma_{2}^{2}\right)$$
where $0 \leq \alpha \leq 1, \sigma_{1}^{2}$ is small and $\sigma_{2}^{2}$ is relatively large. For instance, with $\alpha=$ $0.05$, the finite mixture says that $95 \%$ of the returns follow $N\left(\mu, \sigma_{1}^{2}\right)$ and $5 \%$ follow $N\left(\mu, \sigma_{2}^{2}\right)$. The large value of $\sigma_{2}^{2}$ enables the mixture to put more mass at the tails of its distribution. The low percentage of returns that are from $N\left(\mu, \sigma_{2}^{2}\right)$ says that the majority of the returns follow a simple normal distribution. Advantages of mixtures of normal include that they maintain the tractability of normal, have finite higher order moments, and can capture the excess kurtosis. Yet it is hard to estimate the mixture parameters (e.g., the $\alpha$ in the finite-mixture case).

Figure $1.1$ shows the probability density functions of a finite mixture of normal, Cauchy, and standard normal random variable. The finite mixture of normal is $0.95 N(0,1)+0.05 N(0,16)$ and the density function of Cauchy is
$$f(x)=\frac{1}{\pi\left(1+x^{2}\right)}, \quad-\infty<x<\infty$$
It is seen that Cauchy distribution has fatter tails than the finite mixture of normal, which in turn has fatter tails than the standard normal.

## 统计代写|应用时间序列分析代写applied time series anakysis代考| Stable Distribution

The stable distributions are a natural generalization of normal in that they are stable under addition, which meets the need of continuously compounded returns $r_{t}$. Furthermore, stable distributions are capable of capturing excess kurtosis shown by historical stock returns. However, non-normal stable distributions do not have a finite variance, which is in conflict with most finance theories. In addition, statistical modeling using non-normal stable distributions is difficult. An example of non-normal stable distributions is the Cauchy distribution, which is symmetric with respect to its median, but has infinite variance.

## 统计代写|应用时间序列分析代写applied time series anakysis代考|Multivariate Returns

Let $\boldsymbol{r}{t}=\left(r{1 t}, \ldots, r_{N t}\right)^{\prime}$ be the log returns of $N$ assets at time $t$. The multivariate analyses of Chapters 8 and 9 are concerned with the joint distribution of $\left{\boldsymbol{r}{t}\right}{t=1}^{T}$. This joint distribution can be partitioned in the same way as that of Eq. (1.15). The analysis is then focused on the specification of the conditional distribution function $F\left(\boldsymbol{r}{t} \mid \boldsymbol{r}{t-1}, \ldots, \boldsymbol{r}{1}, \boldsymbol{\theta}\right)$. In particular, how the conditional expectation and conditional covariance matrix of $\boldsymbol{r}{t}$ evolve over time constitute the main subjects of Chapters 8 and $9 .$

The mean vector and covariance matrix of a random vector $X=\left(X_{1}, \ldots, X_{p}\right)$ are defined as
\begin{aligned} E(\boldsymbol{X}) &=\boldsymbol{\mu}{x}=\left[E\left(X{1}\right), \ldots, E\left(X_{p}\right)\right]^{\prime} \ \operatorname{Cov}(\boldsymbol{X}) &=\boldsymbol{\Sigma}{x}=E\left[\left(\boldsymbol{X}-\boldsymbol{\mu}{x}\right)\left(\boldsymbol{X}-\boldsymbol{\mu}{x}\right)^{\prime}\right] \end{aligned} provided that the expectations involved exist. When the data $\left{x{1}, \ldots, x_{T}\right}$ of $X$ are available, the sample mean and covariance matrix are defined as
$$\widehat{\boldsymbol{\mu}}{x}=\frac{1}{T} \sum{t=1}^{T} \boldsymbol{x}{t}, \quad \widehat{\boldsymbol{\Sigma}}{x}=\frac{1}{T} \sum_{t=1}^{T}\left(\boldsymbol{x}{t}-\widehat{\boldsymbol{\mu}}{x}\right)\left(\boldsymbol{x}{t}-\widehat{\boldsymbol{\mu}}{x}\right)^{\prime}$$These sample statistics are consistent estimates of their theoretical counterparts provided that the covariance matrix of $X$ exists. In the finance literature, multivariate normal distribution is often used for the log return $\boldsymbol{r}_{t}$.

## 统计代写|应用时间序列分析代写applied time series anakysis代考|Scale Mixture of Normal Distributions

r吨∼(1−X)ñ(μ,σ12)+Xñ(μ,σ22)

F(X)=1圆周率(1+X2),−∞<X<∞

## 统计代写|应用时间序列分析代写applied time series anakysis代考|Multivariate Returns

μ^X=1吨∑吨=1吨X吨,Σ^X=1吨∑吨=1吨(X吨−μ^X)(X吨−μ^X)′这些样本统计量是对其理论对应物的一致估计，前提是协方差矩阵X存在。在金融文献中，多元正态分布常用于对数回报r吨.

## 广义线性模型代考

statistics-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。