### 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|STATS7062

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• 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|Transformation of Statistics

Often in practical problems, one is interested in a function of parameters for which one has an asymptotically normal statistic. Suppose for instance that we are interested in a cost function depending on the mean $\mu$ of the process: $f(\mu)=$ $\mu^{\top} \mathcal{A} \mu$ where $\mathcal{A}>0$ is given. To estimate $\mu$ we use the asymptotically normal statistic $\bar{x}$. The question is: how does $f(\bar{x})$ behave? More generally, what happens to a statistic $t$ that is asymptotically normal when we transform it by a function $f(t)$ ? The answer is given by the following theorem.

Theorem $4.11$ If $\sqrt{n}(t-\mu) \stackrel{\mathcal{L}}{\longrightarrow} N_{p}(0, \Sigma)$ and if $f=\left(f_{1}, \ldots, f_{q}\right)^{\top}: \mathbb{R}^{p} \rightarrow$ $\mathbb{R}^{q}$ are real valued functions which are differentiable at $\mu \in \mathbb{R}^{p}$, then $f(t)$ is asymptotically normal with mean $f(\mu)$ and covariance $\mathcal{D}^{\top} \Sigma \mathcal{D}$, i.e.
$$\sqrt{n}{f(t)-f(\mu)} \stackrel{\mathcal{L}}{\longrightarrow} N_{q}\left(0, \mathcal{D}^{\top} \Sigma \mathcal{D}\right) \quad \text { for } \quad n \longrightarrow \infty$$
where
$$\mathcal{D}=\left.\left(\frac{\partial f_{j}}{\partial t_{i}}\right)(t)\right|{t=\mu}$$ is the $(p \times q)$ matrix of all partial derivatives. Example $4.20$ We are interested in seeing how $f(\bar{x})=\bar{x}^{\top} \mathcal{A} \bar{x}$ behaves asymptotically with respect to the quadratic cost function of $\mu, f(\mu)=\mu^{\top} \mathcal{A} \mu$, where $\mathcal{A}>0$ $$D=\left.\frac{\partial f(\bar{x})}{\partial \bar{x}}\right|{\bar{x}=\mu}=2 \mathcal{A} \mu$$
By Theorem $4.11$ we have
$$\sqrt{n}\left(\bar{x}^{\top} \mathcal{A} \bar{x}-\mu^{\top} \mathcal{A} \mu\right) \stackrel{\mathcal{L}}{\longrightarrow} N_{1}\left(0,4 \mu^{\top} \mathcal{A} \Sigma \mathcal{A} \mu\right)$$
Example 4.21 Suppose
$$X_{i} \sim(\mu, \Sigma) ; \quad \mu=\left(\begin{array}{l} 0 \ 0 \end{array}\right), \quad \Sigma=\left(\begin{array}{cc} 1 & 0.5 \ 0.5 & 1 \end{array}\right), \quad p=2$$
We have by the CLT (Theorem $4.10$ ) for $n \rightarrow \infty$ that
$$\sqrt{n}(\bar{x}-\mu) \stackrel{\mathcal{L}}{\longrightarrow} N(0, \Sigma) .$$

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Heavy-Tailed Distributions

Heavy-tailed distributions were first introduced by the Italian-born Swiss economist Pareto and extensively studied by Paul Lévy. Although in the beginning these distributions were mainly studied theoretically, nowadays they have found many applications in areas as diverse as finance, medicine, seismology, structural engineering. More concretely, they have been used to model returns of assets in financial markets, stream flow in hydrology, precipitation and hurricane damage in meteorology, earthquake prediction in seismology, pollution, material strength, teletraffic and many others.

A distribution is called heavy-tailed if it has higher probability density in its tail area compared with a normal distribution with same mean $\mu$ and variance $\sigma^{2}$. Figure $4.6$ demonstrates the differences of the pdf curves of a standard Gaussian distribution and a Cauchy distribution with location parameter $\mu=0$ and scale parameter $\sigma=1$. The graphic shows that the probability density of the Cauchy distribution is much higher than that of the Gaussian in the tail part, while in the area around the centre, the probability density of the Cauchy distribution is much lower.

In terms of kurtosis, a heavy-tailed distribution has kurtosis greater than 3 (see Chap. 4 , formula (4.40)), which is called leptokurtic, in contrast to mesokurtic distribution (kurtosis $=3$ ) and platykurtic distribution (kurtosis $<3$ ). Since univariate heavy-tailed distributions serve as basics for their multivariate counterparts and their density properties have been proved useful even in multivariate cases, we will start from introducing some univariate heavy-tailed distributions. Then we will move on to analyse their multivariate counterparts and their tail behaviour.

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

The generalised hyperbolic distribution was introduced by Barndorff-Nielsen and at first applied to model grain size distributions of wind blown sands. Today one of its most important uses is in stock price modelling and market risk measurement. The name of the distribution is derived from the fact that its log-density forms a hyperbola, while the log-density of the normal distribution is a parabola (Fig. 4.7).

The density of a one-dimensional generalised hyperbolic (GH) distribution for $x \in \mathbb{R}$ is
\begin{aligned} &f_{\mathrm{GH}}(x ; \lambda, \alpha, \beta, \delta, \mu) \ &\quad=\frac{\left(\sqrt{\alpha^{2}-\beta^{2}} / \delta\right)^{\lambda}}{\sqrt{2 \pi} K_{\lambda}\left(\delta \sqrt{\alpha^{2}-\beta^{2}}\right)} \frac{K_{\lambda-1 / 2}\left{\alpha \sqrt{\delta^{2}+(x-\mu)^{2}}\right}}{\left.\sqrt{\delta^{2}+(x-\mu)^{2}} / \alpha\right)^{1 / 2-\lambda}} e^{\beta(x-\mu)} \end{aligned}
where $K_{\lambda}$ is a modified Bessel function of the third kind with index $\lambda$
$$K_{\lambda}(x)=\frac{1}{2} \int_{0}^{\infty} y^{\lambda-1} e^{-\frac{1}{2}\left(y+y^{-1}\right)} d y$$
The domain of variation of the parameters is $\mu \in \mathbb{R}$ and
$$\begin{array}{lll} \delta \geq 0,|\beta|<\alpha, & \text { if } \quad \lambda>0 \ \delta>0,|\beta|<\alpha, & \text { if } \quad \lambda=0 \ \delta>0,|\beta| \leq \alpha, & \text { if } \quad \lambda<0 \end{array}$$
The generalised hyperbolic distribution has the following mean and variance
\begin{aligned} \mathrm{E}[X]=& \mu+\frac{\delta \beta}{\sqrt{\alpha^{2}-\beta^{2}}} \frac{K_{\lambda+1}\left(\delta \sqrt{\alpha^{2}-\beta^{2}}\right)}{K_{\lambda}\left(\delta \sqrt{\alpha^{2}-\beta^{2}}\right)} \ \operatorname{Var}[X]=& \delta^{2}\left[\frac{K_{\lambda+1}\left(\delta \sqrt{\alpha^{2}-\beta^{2}}\right)}{\delta \sqrt{\alpha^{2}-\beta^{2}} K_{\lambda}\left(\delta \sqrt{\alpha^{2}-\beta^{2}}\right)}+\frac{\beta^{2}}{\alpha^{2}-\beta^{2}}\left[\frac{K_{\lambda+2}\left(\delta \sqrt{\alpha^{2}-\beta^{2}}\right)}{K_{\lambda}\left(\delta \sqrt{\alpha^{2}-\beta^{2}}\right)}\right.\right.\ &\left.\left.-\left{\frac{K_{\lambda+1}\left(\delta \sqrt{\alpha^{2}-\beta^{2}}\right)}{K_{\lambda}\left(\delta \sqrt{\alpha^{2}-\beta^{2}}\right)}\right}^{2}\right]\right] \end{aligned}
Where $\mu$ and $\delta$ play important roles in the density’s location and scale respectively. With specific values of $\lambda$, we obtain different sub-classes of GH such as hyperbolic (HYP) or normal-inverse Gaussian (NIG) distribution.
For $\lambda=1$ we obtain the hyperbolic distributions (HYP)
$$f_{\mathrm{HYP}}(x ; \alpha, \beta, \delta, \mu)=\frac{\sqrt{\alpha^{2}-\beta^{2}}}{2 \alpha \delta K_{1}\left(\delta \sqrt{\alpha^{2}-\beta^{2}}\right)} e^{\left{-\alpha \sqrt{\delta^{2}+(x-\mu)^{2}}+\beta(x-\mu)\right}}$$
where $x, \mu \in \mathbb{R}, \delta \geq 0$ and $|\beta|<\alpha$. For $\lambda=-1 / 2$ we obtain the NIG distribution
$$f_{\mathrm{NIG}}(x ; \alpha, \beta, \delta, \mu)=\frac{\alpha \delta}{\pi} \frac{K_{1}\left(\alpha \sqrt{\left.\left(\delta^{2}+(x-\mu)^{2}\right)\right)}\right.}{\sqrt{\delta^{2}+(x-\mu)^{2}}} e^{\left{\delta \sqrt{\alpha^{2}-\beta^{2}}+\beta(x-\mu)\right}}$$

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Transformation of Statistics

nF(吨)−F(μ)⟶大号ñq(0,D⊤ΣD) 为了 n⟶∞

D=(∂Fj∂吨一世)(吨)|吨=μ是个(p×q)所有偏导数的矩阵。例子4.20我们有兴趣看看如何F(X¯)=X¯⊤一个X¯关于二次成本函数的行为渐近μ,F(μ)=μ⊤一个μ， 在哪里一个>0

D=∂F(X¯)∂X¯|X¯=μ=2一个μ

n(X¯⊤一个X¯−μ⊤一个μ)⟶大号ñ1(0,4μ⊤一个Σ一个μ)

X一世∼(μ,Σ);μ=(0 0),Σ=(10.5 0.51),p=2

n(X¯−μ)⟶大号ñ(0,Σ).

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

\begin{对齐} &f_{\mathrm{GH}}(x ; \lambda, \alpha, \beta, \delta, \mu) \&\quad=\frac{\left(\sqrt{\alpha^{2 }-\beta^{2}}/\delta\right)^{\lambda}}{\sqrt{2\pi} K_{\lambda}\left(\delta\sqrt{\alpha^{2}-\ beta^{2}}\right)}\frac{K_{\lambda-1/2}\left{\alpha\sqrt{\delta^{2}+(x-\mu)^{2}}\right }}{\left.\sqrt{\delta^{2}+(x-\mu)^{2}}/\alpha\right)^{1/2-\lambda}} e^{\beta(x -\mu)} \end{对齐}\begin{对齐} &f_{\mathrm{GH}}(x ; \lambda, \alpha, \beta, \delta, \mu) \&\quad=\frac{\left(\sqrt{\alpha^{2 }-\beta^{2}}/\delta\right)^{\lambda}}{\sqrt{2\pi} K_{\lambda}\left(\delta\sqrt{\alpha^{2}-\ beta^{2}}\right)}\frac{K_{\lambda-1/2}\left{\alpha\sqrt{\delta^{2}+(x-\mu)^{2}}\right }}{\left.\sqrt{\delta^{2}+(x-\mu)^{2}}/\alpha\right)^{1/2-\lambda}} e^{\beta(x -\mu)} \end{对齐}

ķλ(X)=12∫0∞是λ−1和−12(是+是−1)d是

d≥0,|b|<一个, 如果 λ>0 d>0,|b|<一个, 如果 λ=0 d>0,|b|≤一个, 如果 λ<0

\begin{对齐} \mathrm{E}[X]=& \mu+\frac{\delta \beta}{\sqrt{\alpha^{2}-\beta^{2}}} \frac{K_{\ lambda+1}\left(\delta \sqrt{\alpha^{2}-\beta^{2}}\right)}{K_{\lambda}\left(\delta \sqrt{\alpha^{2} -\beta^{2}}\right)} \ \operatorname{Var}[X]=& \delta^{2}\left[\frac{K_{\lambda+1}\left(\delta \sqrt{ \alpha^{2}-\beta^{2}}\right)}{\delta \sqrt{\alpha^{2}-\beta^{2}} K_{\lambda}\left(\delta \sqrt {\alpha^{2}-\beta^{2}}\right)}+\frac{\beta^{2}}{\alpha^{2}-\beta^{2}}\left[\frac {K_{\lambda+2}\left(\delta \sqrt{\alpha^{2}-\beta^{2}}\right)}{K_{\lambda}\left(\delta \sqrt{\alpha ^{2}-\beta^{2}}\right)}\right.\right.\ &\left.\left.-\left{\frac{K_{\lambda+1}\left(\delta \ sqrt{\alpha^{2}-\beta^{2}}\right)}{K_{\lambda}\left(\delta \sqrt{\alpha^{2}-\beta^{2}}\right )}\right}^{2}\right]\right] \end{对齐}\begin{对齐} \mathrm{E}[X]=& \mu+\frac{\delta \beta}{\sqrt{\alpha^{2}-\beta^{2}}} \frac{K_{\ lambda+1}\left(\delta \sqrt{\alpha^{2}-\beta^{2}}\right)}{K_{\lambda}\left(\delta \sqrt{\alpha^{2} -\beta^{2}}\right)} \ \operatorname{Var}[X]=& \delta^{2}\left[\frac{K_{\lambda+1}\left(\delta \sqrt{ \alpha^{2}-\beta^{2}}\right)}{\delta \sqrt{\alpha^{2}-\beta^{2}} K_{\lambda}\left(\delta \sqrt {\alpha^{2}-\beta^{2}}\right)}+\frac{\beta^{2}}{\alpha^{2}-\beta^{2}}\left[\frac {K_{\lambda+2}\left(\delta \sqrt{\alpha^{2}-\beta^{2}}\right)}{K_{\lambda}\left(\delta \sqrt{\alpha ^{2}-\beta^{2}}\right)}\right.\right.\ &\left.\left.-\left{\frac{K_{\lambda+1}\left(\delta \ sqrt{\alpha^{2}-\beta^{2}}\right)}{K_{\lambda}\left(\delta \sqrt{\alpha^{2}-\beta^{2}}\right )}\right}^{2}\right]\right] \end{对齐}

f_{\mathrm{HYP}}(x ; \alpha, \beta, \delta, \mu)=\frac{\sqrt{\alpha^{2}-\beta^{2}}}{2 \alpha \ delta K_{1}\left(\delta \sqrt{\alpha^{2}-\beta^{2}}\right)} e^{\left{-\alpha \sqrt{\delta^{2}+ (x-\mu)^{2}}+\beta(x-\mu)\right}}f_{\mathrm{HYP}}(x ; \alpha, \beta, \delta, \mu)=\frac{\sqrt{\alpha^{2}-\beta^{2}}}{2 \alpha \ delta K_{1}\left(\delta \sqrt{\alpha^{2}-\beta^{2}}\right)} e^{\left{-\alpha \sqrt{\delta^{2}+ (x-\mu)^{2}}+\beta(x-\mu)\right}}

f_{\mathrm{NIG}}(x ; \alpha, \beta, \delta, \mu)=\frac{\alpha \delta}{\pi} \frac{K_{1}\left(\alpha \sqrt {\left.\left(\delta^{2}+(x-\mu)^{2}\right)\right)}\right.}{\sqrt{\delta^{2}+(x-\ mu)^{2}}} e^{\left{\delta \sqrt{\alpha^{2}-\beta^{2}}+\beta(x-\mu)\right}}f_{\mathrm{NIG}}(x ; \alpha, \beta, \delta, \mu)=\frac{\alpha \delta}{\pi} \frac{K_{1}\left(\alpha \sqrt {\left.\left(\delta^{2}+(x-\mu)^{2}\right)\right)}\right.}{\sqrt{\delta^{2}+(x-\ mu)^{2}}} e^{\left{\delta \sqrt{\alpha^{2}-\beta^{2}}+\beta(x-\mu)\right}}

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