### 统计代写|金融统计代写Financial Statistics代考|ST326

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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|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
$$f(\theta, \rho, a, \mu)=\frac{1}{2 \pi}\left[1+2 \sum_{p=1}^{\infty} \rho^{p^{n}} \cos p(\theta-\mu)\right] \quad 0<\rho \leq 1,0<\alpha \leq 2,0<\mu \leq 2 \pi$$
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 $\quad a_{p}=E \cos p(\theta-\mu)$ and $\beta_{p}=E \sin p(\theta-\mu)$
Furthermore, from Jammalamadaka and SenGupta (2001), it is known that for, the p.d.f given by Equation (1),
$$\psi_{\theta}(p)=\rho^{p^{n}}$$
Hence, $E \cos p(\theta-\mu)=\rho^{p^{*}}$ and $E \sin p(\theta-\mu)=0$
We define
$$C_{1}=\frac{1}{m} \sum_{i=1}^{m} \cos \theta_{i}, \quad C_{2}=\frac{1}{m} \sum_{i=1}^{m} \cos 2 \theta_{i}, \quad S_{1}=\frac{1}{m} \sum_{i=1}^{m} \sin \theta_{i}$$
and $\quad \bar{S}{2}=\frac{1}{m} \sum{i=1}^{m} \sin 2 \theta_{i}$
Then, we note that $\bar{R}{1}=\sqrt{{\overline{C{1}}}^{2}+{\overline{S_{1}}}^{2}}$ and $\bar{R}{2}=\sqrt{{\overline{C{2}}}^{2}+\bar{S}_{2}^{2}}$

By the method of trigonometric moments estimation, equating $K_{1}$ and $R_{2}$ to the corresponding functions of the theoretical trigonometric moments, we get the estimator of index parameter $\alpha$ as (see SenGupta (1996)):
$$\hat{k}=\frac{1}{\ln 2} \ln \frac{\ln \bar{R}{2}}{\ln \bar{R}{1}}$$
Then, we define $\bar{R}{j}=\frac{1}{m} \sum{i=1}^{m} \cos j\left(\theta_{i}-\bar{\theta}\right), j=1,2$ and $\bar{\theta}$ is the mean direction given by $\bar{\theta}=\arctan \left(\frac{\xi_{1}}{C_{1}}\right)$. Note that $K_{1} \equiv R$.
We consider two special cases.

## 统计代写|金融统计代写Financial Statistics代考|Improvement Over the Moment Estimator

The moment estimator need not always remain in the support of the true parameter $\alpha($ that is $(0,2])$. Hence, the moment estimators proposed above do not need to be proper estimators of $\alpha$. A modified estimator free from this defect is given by
\begin{aligned} \hat{a^{*}} &=\hat{u} \quad \text { if } 0<\hat{u}<2 \ &=2 \quad \text { if } \hat{a} \geq 2 \end{aligned}
(since support of $a$ excludes non-positive values).
Thus, the density function of $\hat{\alpha}$ * is given by
\begin{aligned} g\left(\hat{a^{}}\right) &=\frac{P[\hat{a}<2]}{P[\hat{a} \geq 0]} \quad ; 0<\hat{a^{}}<2 \equiv-\infty<\hat{a}<2 \ &=P\left[\hat{a^{}}=2\right] \quad ; \hat{a^{}}=2 \equiv \hat{k} \geq 2 \ &=\frac{P[\hat{a} \geq 2]}{P[\hat{a} \geq 0]} \quad ; \hat{a^{}}=2 \equiv \hat{a} \geq 2 \end{aligned} where $f(\hat{\alpha})$ is the density function of $\hat{k} \sim N(a, \gamma / \Sigma \gamma / m)$. Therefore, $g\left(\hat{\alpha}^{}\right)=\frac{\Phi\left(\frac{2-a}{\sqrt{I^{12} / m}}\right)}{1-\Phi\left(\frac{-a}{\sqrt{I^{2}-\sqrt{y}}}\right)} \quad ; 0<\hat{a^{}}<2 \equiv-\infty$ $=1-\frac{\Phi\left(\frac{2-a}{\sqrt{x^{12} y^{/ m}}}\right)}{\Phi\left(\frac{a}{\sqrt{I^{2}} y^{/ m}}\right)} ; \hat{x^{}}=2 \equiv \hat{k} \geq 2$
Thus, we get $g\left(\hat{\alpha}^{}\right)$ as a mixed distribution of one atomic mass function and a continuous function. 4.1. Special Case 1: $\mu=0, \sigma=1$ 4.2. Special Case 2 : $\mu=0, \sigma$ Lnknown Similar modifications can be made for the estimator ${\hat{\alpha_{2}}}^{\text {. }}$. Let it be denoted by $\hat{\alpha_{2}^{}}$.

## 统计代写|金融统计代写Financial Statistics代考|Derivation of the Asymptotic Distribution of the Modified Truncated Estimators

Now, using the asymptotic normal distribution of $\tilde{a}$, we can derive the same results for the modified truncated estimator of the index parameter $\alpha$ (given as below) as we have done for the method of moment estimator of $\alpha$.
The mean of $\hat{a^{}}$ is given by $$E\left(\hat{\alpha}^{}\right)=0 . P(\hat{a}<0)+\int_{0}^{2} \hat{\alpha} f(\hat{u}) d \hat{u}+2 \cdot P(\hat{a}>2)$$
where $\sqrt{m}(\hat{\alpha}-\alpha) \rightarrow \mathrm{N}\left(0, \underline{y}^{\prime} \underline{y}\right)$ asymptotically (as noted above) and $\mathrm{f}(\hat{\alpha})=$ probability density function of $\hat{a}$.
The above is equivalent to $\tau=\frac{\frac{\pi-\pi}{\sqrt{y^{2} \gamma^{\prime m}}}}{}$
Let $\phi(\tau)$ and $\Phi(\tau)$ denote the p.d.f. and c.d.f. of $\tau$, respectively.
Let $\sigma=\sqrt{\frac{m^{2}}{m}}$. Then, we get,
\begin{aligned} &E\left(\hat{\alpha}^{}\right)=a P\left(\tau}\right)+\int_{a^{}}^{b^{}}(\tau \sigma+\alpha) \phi(\tau) d \tau+b P\left(\tau>b^{}\right) \ &\Rightarrow E\left(\hat{\alpha}^{}\right)=\sigma\left[\left{\phi\left(a^{}\right)-\phi\left(b^{}\right)\right}\right]+\alpha\left[\Phi\left(b^{}\right)-\Phi\left(a^{}\right)\right] \end{aligned}
$=\alpha$
since $\left[\Phi\left(b^{}\right)-\Phi\left(a^{}\right)\right] \rightarrow 1, b\left[1-\Phi\left(b^{}\right)\right] \rightarrow 0$ and $\sigma \rightarrow 0$ as $m \rightarrow$ infinity where $a^{}=\frac{-a}{\sqrt{\frac{L^{2}}{m}}} \quad$ and $b^{}=\frac{2-\alpha}{\sqrt{\frac{\pi^{2}-y}{m}}}$ $E\left(\hat{\alpha}^{2}\right)=0^{2} \cdot P(\hat{\alpha}<0)+\int_{0}^{2} \hat{\alpha}^{2} f(\hat{\alpha}) \mathrm{d} \hat{\alpha}+4 \cdot P(\hat{\alpha}>2)$ $$\Rightarrow E\left(\hat{a}^{2}\right)=\sigma^{2}\left[\left{a^{} \phi\left(a^{}\right)-b^{} \phi\left(b^{}\right)+\Phi\left(b^{}\right)-\Phi\left(a^{}\right)\right}\right]+\alpha^{2}\left{\Phi\left(b^{}\right)-\Phi\left(a^{}\right)\right}+2 \alpha \sigma\left{\phi\left(a^{}\right)-\right.$$
$\left.\phi\left(b^{}\right)\right}$ since $b^{2} .\left[1-\Phi\left(b^{\prime}\right)\right] \rightarrow 0$ as $m \rightarrow$ infinity The asymptotic variance of $\hat{\alpha^{}}$ is given by
$$V\left(\hat{\alpha^{}}\right)=E\left({\hat{a^{}}}^{2}\right)-\left[E\left(\hat{a}^{}\right)\right]^{2}$$ Similarly, the mean of $\hat{x_{1}^{}}$ is given by
$$\begin{gathered} E\left(\hat{a}{1}^{*}\right)=\frac{\left.\sigma \frac{\left(\partial y\left(T{m}^{\prime}\right)\right.}{d}\right)}{\sqrt{m}}\left[\left{\phi\left(a^{\prime}\right)-\phi\left(b^{\prime}\right)\right}\right]+\alpha\left[\Phi\left(b^{\prime}\right)-\Phi\left(a^{\prime}\right)\right] \text { since } b\left[1-\Phi\left(b^{\prime}\right)\right] \rightarrow 0 \text { as } m \rightarrow \text { infinity } \ E\left(\hat{a}{1}^{2}\right)=\frac{\left.\sigma^{2} \frac{\partial \partial\left(T{m}^{\prime}\right)}{\partial \mu^{\prime}}\right)^{2}}{m}\left[\left{a^{\prime} \phi\left(a^{\prime}\right)-b^{\prime} \phi\left(b^{\prime}\right)+\Phi\left(b^{\prime}\right)-\Phi\left(a^{\prime}\right)\right}\right]+\alpha^{2}\left{\Phi\left(b^{\prime}\right)-\Phi\left(a^{\prime}\right)\right}+ \ 2 \alpha \frac{\left.\sigma \frac{\left(d x\left(J_{m}^{\prime}\right)\right.}{d m}\right)}{\sqrt{m}}\left{\phi\left(a^{\prime}\right)-\phi\left(b^{\prime}\right)\right} \text { since } b^{2} \cdot\left[1-\Phi\left(b^{\prime}\right)\right] \rightarrow 0 \text { as } m \rightarrow \text { infinity } \end{gathered}$$

## 统计代写|金融统计代写Financial Statistics代考|The Trigonometric Moment Estimator

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

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

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

$$\psi_{\theta}(p)=\rho^{p^{n}}$$

$$C_{1}=\frac{1}{m} \sum_{i=1}^{m} \cos \theta_{i}, \quad C_{2}=\frac{1}{m} \sum_{i=1}^{m} \cos 2 \theta_{i}, \quad S_{1}=\frac{1}{m} \sum_{i=1}^{m} \sin \theta_{i}$$

$$\hat{k}=\frac{1}{\ln 2} \ln \frac{\ln \bar{R} 2}{\ln \bar{R} 1}$$

## 统计代写|金融统计代写Financial Statistics代考|Improvement Over the Moment Estimator

$$\hat{a^{*}}=\hat{u} \quad \text { if } 0<\hat{u}<2 \quad=2 \quad \text { if } \hat{a} \geq 2$$
(由于支持 $a$ 不包括非正值)。

$$g(\hat{a})=\frac{P[\hat{a}<2]}{P[\hat{a} \geq 0]} \quad ; 0<\hat{a}<2 \equiv-\infty<\hat{a}<2 \quad=P[\hat{a}=2] \quad ; \hat{a}=2 \equiv \hat{k} \geq 2=\frac{P[\hat{a} \geq 2]}{P[\hat{a} \geq 0]} \quad ; \hat{a}=$$

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