Posted on Categories:Multivariate Statistical Analysis, 多元统计分析, 统计代写, 统计代考

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## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Multivariate t-distribution

If $X$ and $Y$ are independent and distributed as $N_p(\mu, \Sigma)$ and $\mathcal{X}_n^2$ respectively, and $X \sqrt{n / Y}=$ $t-\mu$, then the pdf of $t$ is given by
$$f_t(t ; n, \Sigma, \mu)=\frac{\Gamma{(n+p) / 2}}{\Gamma(n / 2) n^{p / 2} \pi^{p / 2}|\Sigma|^{1 / 2}\left{1+\frac{1}{n}(t-\mu)^{\top} \Sigma^{-1}(t-\mu)\right}^{(n+p) / 2}}$$
The distribution of $t$ is the noncentral $t$-distribution with $n$ degrees of freedom and the noncentrality parameter $\mu$, Giri (1996).
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
\begin{aligned} f_{\text {MLaplaced }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{\text {Maplace }d}(x, m, \Sigma) & =\int_0^{\infty} G\left(z^{-\frac{1}{2}} x-z^{\frac{1}{2}} m\right) e^{-z} d z \end{aligned} the pdf can also be described as \begin{aligned} f{\text {MLaplace }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{\lambda}{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
$$K_\lambda(x)=\frac{1}{2}\left(\frac{x}{2}\right)^\lambda \int_0^{\infty} t^{-\lambda-1} e^{-t-\frac{x^2}{4 t}} d t, \quad x>0$$
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|Multivariate Mixture Model

A multivariate mixture model comprises multivariate distributions, e.g. the pdf of a multivariate Gaussian distribution can be written as
$$f(x)=\sum_{l=1}^n \frac{w_l}{\left|2 \pi \Sigma_l\right|^{\frac{1}{2}}} e^{-\frac{1}{2}\left(x-\mu_l\right)^{\top} \Sigma^{-1}\left(x-\mu_l\right)}$$
Generalized Hyperbolic Distribution
The GH distribution has an exponential decaying speed
$$f_{G H}(x ; \lambda, \alpha, \beta, \delta, \mu=0) \sim x^{\lambda-1} e^{-(\alpha-\beta) x} \quad \text { as } \quad x \rightarrow \infty$$
Figure 4.14 illustrates the tail behavior of GH distributions with different value of $\lambda$ with $\alpha=1, \beta=0, \delta=1, \mu=0$. The left panel contains part of pdf curves of GH distribution and the right panel demonstrates the approximation by the function mentioned above. It is clear that among the four distributions, GH with $\lambda=1.5$ has the lowest decaying speed, while NIG decays fastest.

In Figure 4.15, Chen, Härdle and Jeong (2005), four distributions and especially their tailbehavior are compared. In order to keep the comparability of these distributions, we specified the means to 0 and standardized the variances to 1 . Furthermore we used one important subclass of the GH distribution: the normal-inverse Gaussian (NIG) distribution with $\lambda=-\frac{1}{2}$ introduced above. On the left panel, the complete forms of these distributions are revealed. The Cauchy (dots) distribution has the lowest peak and the fattest tails. In other words, it has the flattest distribution. The NIG distribution decays second fast in the tails although it has the highest peak, which is more clearly displayed on the right panel.

## 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。