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## 数据科学代写|金融统计代写Financial Statistics代考|The Trigonometric Moment Estimator

The regular symmetric stable distribution is defined through its characteristic function given by
$$\varphi(t)=\exp \left(i t \mu-|\sigma t|^{a}\right)$$
where $\mu$ is the location parameter; $\sigma$ is the scale parameter, which we take as 1 ; and $\alpha$ is the index or shape parameter of the distribution. Here, without loss of generality, we take $\mu=0$.

From the stable distribution, we can obtain the wrapped stable distribution (the process of wrapping explained in Jammalamadaka and SenGupta (2001)). Suppose $\theta_{1}, \theta_{2}, \ldots, \theta_{m}$ is a random sample of size $m$ drawn from the wrapped stable (given in Jammalamadaka and SenGupta (2001)) distribution whose probability density function is given by
$$f(\theta, \rho, \alpha, \mu)=\frac{1}{2 \pi}\left[1+2 \sum_{p=1}^{\infty} \rho^{p^{\alpha}} \cos p(\theta-\mu)\right] \quad 0<\rho \leq 1,0<\alpha \leq 2,0<\mu \leq 2 \pi$$
It is known in general from Jammalamadaka and SenGupta (2001) that the characteristic function of $\theta$ at the integer $p$ is defined as,
$$\psi_{\theta}(p)=E[\exp (i p(\theta-\mu))]=\alpha_{p}+i \beta_{p}$$
where $\quad \alpha_{p}=E \cos p(\theta-\mu) \quad$ and $\quad \beta_{p}=E \sin p(\theta-\mu)$

## 数据科学代写|金融统计代写Financial Statistics代考|Derivation of the Asymptotic Distribution of the Moment Estimator

Lemma $1 .$
$$\sqrt{m}\left(T_{m}-\mu\right) \stackrel{L}{\rightarrow} N_{4}(0, \Sigma)$$
where $T_{m}=\left(\bar{C}{1}, \overline{C{2}}, \overline{S_{1}}, \overline{S_{2}}\right)^{\prime}$,
$\mu$ is the mean vector given by
$$\mu=\left(\rho \cos \mu_{0}, \rho^{2^{\alpha}} \cos 2 \mu_{0}, \rho \sin \mu_{0}, \rho^{2^{a}} \sin 2 \mu_{0}\right)^{\prime}$$
and $\Sigma$ is the dispersion matrix given by

$$\Sigma=\left(\begin{array}{cccc} A & B & C & D \ B & E & F & G \ C & F & H & I \ D & G & I & J \end{array}\right)$$
where
$A=\frac{\rho^{2 \pi} \cos 2 \mu_{0}+1-2 \rho^{2} \cos ^{2} \mu_{0}}{2}$
$B=\frac{\rho \cos \mu_{0}+\rho^{3 \pi} \cos 3 \mu_{0}-22^{22^{\alpha}+1} \cos \mu_{0} \cos 2 \mu_{0}}{2}$
$C=\frac{\rho^{2 \pi} \sin 2 \mu_{0}-2 \rho^{2} \cos \mu_{0} \sin \mu_{0}}{2}$
$D=\frac{\rho^{3 \pi} \sin 3 \mu_{0}+\rho \sin \mu_{0}-22^{22^{\alpha}+1} \cos \mu_{0} \sin 2 \mu_{0}}{\alpha^{2}}$
$E=\frac{\rho^{4^{a}} \cos 4 \mu_{0}+1-2\left(\rho^{2 \alpha^{2}}\right)^{2} \cos ^{2} 2 \mu_{0}}{2}$
$F=\frac{\rho^{3^{\pi}} \sin 3 \mu_{0}-\rho \sin \mu_{0}-2 \rho^{2^{\alpha}}+1 \cos 2 \mu_{0} \sin \mu_{0}}{2}$
$G=\frac{\rho^{\rho^{a x}} \sin 4 \mu_{0}-2\left(\rho^{2 x}\right)^{2} \cos 2 \mu_{0} \sin 2 \mu_{0}}{2}$
$H=\frac{1-\rho^{2 \alpha} \cos 2 \mu_{0}-2 \rho^{2} \sin ^{2} \mu_{0}}{2}$
$I=\frac{\rho \cos \mu_{0}-\rho^{3^{\alpha}} \cos 3 \mu_{0}-2 \rho^{2^{\alpha}}+1 \sin \mu_{0} \sin 2 \mu_{0}}{2}$
$J=\frac{1-\rho^{4^{\alpha}} \cos 4 \mu_{0}-2\left(\rho^{2^{2}}\right)^{2} \sin ^{2} 2 \mu_{0}}{2}$
Proof. The derivations for the proof are given in Appendix A.

## 数据科学代写|金融统计代写Financial Statistics代考|The Trigonometric Moment Estimator

$$\varphi(t)=\exp \left(i t \mu-|\sigma t|^{a}\right)$$

$$f(\theta, \rho, \alpha, \mu)=\frac{1}{2 \pi}\left[1+2 \sum_{p=1}^{\infty} \rho^{p^{\alpha}} \cos p(\theta-\mu)\right] \quad 0<\rho \leq 1,0<\alpha \leq 2,0<\mu \leq 2 \pi$$

$$\psi_{\theta}(p)=E[\exp (i p(\theta-\mu))]=\alpha_{p}+i \beta_{p}$$

## 数据科学代写|金融统计代写Financial Statistics代考|Derivation of the Asymptotic Distribution of the Moment Estimator

$$\sqrt{m}\left(T_{m}-\mu\right) \stackrel{L}{\rightarrow} N_{4}(0, \Sigma)$$

$\mu$ 是由下式给出的平均向量
$$\mu=\left(\rho \cos \mu_{0}, \rho^{2^{\alpha}} \cos 2 \mu_{0}, \rho \sin \mu_{0}, \rho^{2^{a}} \sin 2 \mu_{0}\right)^{\prime}$$

$A=\frac{\rho^{2 \pi} \cos 2 \mu_{0}+1-2 \rho^{2} \cos ^{2} \mu_{0}}{2}$
$B=\frac{\rho \cos \mu_{0}+\rho^{3 \pi} \cos 3 \mu_{0}-22^{2 \alpha^{2}+1} \cos \mu_{0} \cos 2 \mu_{0}}{2}$
\begin{aligned} &C=\frac{\rho^{2 \pi} \sin 2 \mu_{0}-2 \rho^{2} \cos \mu_{0} \sin \mu_{0}}{2} \ &D=\frac{\rho^{3 \pi} \sin 3 \mu_{0}+\rho \sin \mu_{0}-22^{22^{\alpha}+1} \cos \mu_{0} \sin 2 \mu_{0}}{\alpha^{2}} \ &E=\frac{\rho^{4 a} \cos 4 \mu_{0}+1-2\left(\rho^{22^{2}}\right)^{2} \cos ^{2} 2 \mu_{0}}{2} \ &F=\frac{\rho^{3 \pi} \sin 3 \mu_{0}-\rho \sin \mu_{0}-2 \rho^{22^{\alpha}}+1 \cos 2 \mu 0 \sin \mu_{0}}{2} \ &G=\frac{\rho^{\rho^{\alpha x}} \sin 4 \mu_{0}-2\left(\rho^{2 x}\right)^{2} \cos 2 \mu_{0} \sin 2 \mu_{0}}{2} \ &H=\frac{1-\rho^{2 \alpha} \cos 2 \mu_{0}-2 \rho^{2} \sin ^{2} \mu_{0}}{2} \ &I=\frac{\rho \cos \mu_{0}-\rho^{3 \alpha} \cos 3 \mu_{0}-2 \rho^{2 \alpha}+1 \sin \mu_{0} \sin 2 \mu_{0}}{2} \ &J=\frac{1-\rho^{\rho^{\alpha}} \cos 4 \mu_{0}-2\left(\rho^{2}\right)^{2} \sin ^{2} 2 \mu_{0}}{2} \end{aligned}

## 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。