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

#### Doug I. Jones

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

Mixture modeling concerns modeling a statistical distribution by a mixture (or weighted sum) of different distributions. For many choices of component density functions, the mixture model can approximate any continuous density to arbitrary accuracy, provided that the number of component density functions is sufficiently large and the parameters of the model are chosen correctly. The pdf of a mixture distribution consists of $n$ distributions and can be written as:
$$f(x)=\sum_{l=1}^n w_l p_l(x)$$
under the constraints:
$$\begin{gathered} 0 \leq w_l \leq 1 \ \sum_{l=1}^n w_l=1 \ \int p_l(x) d x=1 \end{gathered}$$
where $p_l(x)$ is the pdf of the $l$ ‘th component density and $w_l$ is a weight. The mean, variance, skewness and kurtosis of a mixture are
\begin{aligned} \mu= & \sum_{l=1}^n w_l \mu_l \ \sigma^2= & \sum_{l=1}^n w_l\left{\sigma_l^2+\left(\mu_l-\mu\right)^2\right} \ \text { Skewness }= & \sum_{l=1}^n w_l\left{\left(\frac{\sigma_l}{\sigma}\right)^3 S K_l+\frac{3 \sigma_l^2\left(\mu_l-\mu\right)}{\sigma^3}+\left(\frac{\mu_l-\mu}{\sigma}\right)^3\right} \ \text { Kurtosis }= & \sum_{l=1}^n w_l\left{\left(\frac{\sigma_l}{\sigma}\right)^4 K_l+\frac{6\left(\mu_l-\mu\right)^2 \sigma_l^2}{\sigma^4}+\frac{4\left(\mu_l-\mu\right) \sigma_l^3}{\sigma^4} S K_l\right. \ & \left.+\left(\frac{\mu_l-\mu}{\sigma}\right)^4\right}, \end{aligned}
where $\mu_l, \sigma_l, S K_l$ and $K_l$ are respectively mean, variance, skewness and kurtosis of $l$ ‘th distribution.

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

The multivariate Generalized Hyperbolic Distribution $\left(G H_d\right)$ has the following pdf
\begin{aligned} & f_{G H_d}(x ; \lambda, \alpha, \beta, \delta, \Delta, \mu)=a_d \frac{K_{\lambda-\frac{d}{2}}\left{\alpha \sqrt{\delta^2+(x-\mu)^{\top} \Delta^{-1}(x-\mu)}\right}}{\left{\alpha^{-1} \sqrt{\delta^2+(x-\mu)^{\top} \Delta^{-1}(x-\mu)}\right}^{\frac{d}{2}-\lambda}} e^{\beta^{\top}(x-\mu)} \ & a_d=a_d(\lambda, \alpha, \beta, \delta, \Delta)=\frac{\left(\sqrt{\alpha^2-\beta^{\top} \Delta \beta} / \delta\right)^\lambda}{(2 \pi)^{\frac{d}{2}} K_\lambda\left(\delta \sqrt{\alpha^2-\beta^{\top} \Delta \beta}\right.}, \end{aligned}
and characteristic function

\begin{aligned} \phi(t)= & \left(\frac{\alpha^2-\beta^{\top} \Delta \beta}{\alpha^2-\beta^{\top} \Delta \beta+\frac{1}{2} t^{\top} \Delta t-i \beta^{\top} \Delta t}\right)^{\frac{\lambda}{2}} \ & \times \frac{K_\lambda\left(\delta \sqrt{\alpha^2-\beta^{\top} \Delta \beta^{\top}+\frac{1}{2} t^{\top} \Delta t-i \beta^{\top} \Delta t}\right)}{K_\lambda\left(\delta \sqrt{\alpha^2-\beta^{\top} \Delta \beta^{\top}}\right)} \end{aligned}
These parameters have the following domain of variation:
$$\begin{array}{ll} \lambda \in \mathbb{R}, & \beta, \mu \in \mathbb{R}^d \ \delta>0, & \alpha>\beta^{\top} \Delta \beta \ \Delta \in \mathbb{R}^{d \times d} & \text { positive definite matrix } \ |\Delta|=1 & \end{array}$$
For $\lambda=\frac{d+1}{2}$ we obtain the multivariate hyperbolic (HYP) distribution; for $\lambda=-\frac{1}{2}$ we get the multivariate normal inverse Gaussian (NIG) distribution.
Blæsild and Jensen (1981) introduced a second parameterization $(\zeta, \Pi, \Sigma)$, where
\begin{aligned} \zeta & =\delta \sqrt{\alpha^2-\beta^{\top} \Delta \beta} \ \Pi & =\beta \sqrt{\frac{\Delta}{\alpha^2-\beta^{\top} \Delta \beta}} \ \Sigma & =\delta^2 \Delta \end{aligned}

# 多元统计分析代考

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

$$f(x)=\sum_{l=1}^n w_l p_l(x)$$

$$\begin{gathered} 0 \leq w_l \leq 1 \ \sum_{l=1}^n w_l=1 \ \int p_l(x) d x=1 \end{gathered}$$

\begin{aligned} \mu= & \sum_{l=1}^n w_l \mu_l \ \sigma^2= & \sum_{l=1}^n w_l\left{\sigma_l^2+\left(\mu_l-\mu\right)^2\right} \ \text { Skewness }= & \sum_{l=1}^n w_l\left{\left(\frac{\sigma_l}{\sigma}\right)^3 S K_l+\frac{3 \sigma_l^2\left(\mu_l-\mu\right)}{\sigma^3}+\left(\frac{\mu_l-\mu}{\sigma}\right)^3\right} \ \text { Kurtosis }= & \sum_{l=1}^n w_l\left{\left(\frac{\sigma_l}{\sigma}\right)^4 K_l+\frac{6\left(\mu_l-\mu\right)^2 \sigma_l^2}{\sigma^4}+\frac{4\left(\mu_l-\mu\right) \sigma_l^3}{\sigma^4} S K_l\right. \ & \left.+\left(\frac{\mu_l-\mu}{\sigma}\right)^4\right}, \end{aligned}

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

\begin{aligned} & f_{G H_d}(x ; \lambda, \alpha, \beta, \delta, \Delta, \mu)=a_d \frac{K_{\lambda-\frac{d}{2}}\left{\alpha \sqrt{\delta^2+(x-\mu)^{\top} \Delta^{-1}(x-\mu)}\right}}{\left{\alpha^{-1} \sqrt{\delta^2+(x-\mu)^{\top} \Delta^{-1}(x-\mu)}\right}^{\frac{d}{2}-\lambda}} e^{\beta^{\top}(x-\mu)} \ & a_d=a_d(\lambda, \alpha, \beta, \delta, \Delta)=\frac{\left(\sqrt{\alpha^2-\beta^{\top} \Delta \beta} / \delta\right)^\lambda}{(2 \pi)^{\frac{d}{2}} K_\lambda\left(\delta \sqrt{\alpha^2-\beta^{\top} \Delta \beta}\right.}, \end{aligned}

\begin{aligned} \phi(t)= & \left(\frac{\alpha^2-\beta^{\top} \Delta \beta}{\alpha^2-\beta^{\top} \Delta \beta+\frac{1}{2} t^{\top} \Delta t-i \beta^{\top} \Delta t}\right)^{\frac{\lambda}{2}} \ & \times \frac{K_\lambda\left(\delta \sqrt{\alpha^2-\beta^{\top} \Delta \beta^{\top}+\frac{1}{2} t^{\top} \Delta t-i \beta^{\top} \Delta t}\right)}{K_\lambda\left(\delta \sqrt{\alpha^2-\beta^{\top} \Delta \beta^{\top}}\right)} \end{aligned}

$$\begin{array}{ll} \lambda \in \mathbb{R}, & \beta, \mu \in \mathbb{R}^d \ \delta>0, & \alpha>\beta^{\top} \Delta \beta \ \Delta \in \mathbb{R}^{d \times d} & \text { positive definite matrix } \ |\Delta|=1 & \end{array}$$

b æsild和Jensen(1981)引入了第二个参数化$(\zeta, \Pi, \Sigma)$，其中
\begin{aligned} \zeta & =\delta \sqrt{\alpha^2-\beta^{\top} \Delta \beta} \ \Pi & =\beta \sqrt{\frac{\Delta}{\alpha^2-\beta^{\top} \Delta \beta}} \ \Sigma & =\delta^2 \Delta \end{aligned}

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