统计代写|金融统计代写Financial Statistics代考|AEM4070

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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|Applications

8.1. Inference on the Gold Price Data (In US Dollars) (1980-2013)
Gold price data, say $x_{t}$, were collected per ounce in US dollars over the years $1980-2013 .$ These were transformed as $z_{t}=100\left(\ln \left(x_{t}\right)-\ln \left(x_{t-1}\right)\right)$, which were then “wrapped” to obtain $\theta_{t}=z_{t} \bmod 2 \pi$ and finally transformed to $\hat{\theta}=\left(\theta_{t}-\bar{\theta}\right) \bmod 2 \pi$, where $\bar{\theta}$ denotes the mean direction of $\theta_{t}$ and $\hat{\theta}$ denotes the variable thetamod as used in the graphs. The Durbin-Watson test performed on the log ratio transformed data shows that the autocorrelation is zero. The test statistic of Watson’s goodness of fit Jammalamadaka and SenGupta (2001) for wrapped stable distribution was obtained as $0.01632691$ and the corresponding P-value was obtained as $0.9970284$, which is greater than $0.05$, indicating that the wrapped stable distribution fits the transformed gold price data (in US dollars). The modified truncated estimate $\hat{\alpha}_{1}^{*}$ is $0.3752206$ while the estimate by characteristic function method is $0.401409$. The value of the objective function using the characteristic function estimate is $2.218941$ while that using our modified truncated estimate is $2.411018$.
8.2. Inference on the Silver Price Data (In US Dollars) (1980-2013)
Data on the price of silver in US dollars collected per ounce over the same time period also underwent the same transformation. The Durbin-Watson test performed on the log ratio transformed data shows that the autocorrelation is zero. Here, the Watson’s goodness of fit test for wrapped stable distribution was also performed and the value of the statistic was obtained as $0.02530653$ and the corresponding $p$-value is $0.9639666$, which is greater than $0.05$, indicating that the wrapped stable distribution also fits the transformed silver price data (in US dollars). The modified truncated estimate of the index parameter $\alpha$ is $0.4112475$ while the estimate by characteristic function method is $0.644846 .$ The value of the objective function using the characteristic function estimate is $2.234203$ while that using our modified truncated estimate is $2.234432$.

统计代写|金融统计代写Financial Statistics代考|Findings and Concluding Remarks

It can be observed from Table 1 that the asymptotic variance of the untruncated estimator is reduced for the corresponding truncated estimator, indicating the efficiency of the truncated estimator.
It can also be noted from Table 2 that, for $\alpha=1.01$, the RMSE of the modified truncated estimator is less than that of the Hill estimator when the sample is relocated by three different relocations, viz. true mean $=0$, sample mean, and sample median, for higher values of the concentration parameter $\rho=0.5,0.6,0.8$, and $0.9$ for sample sizes $n=100,250,500$, and 1000 and for $\rho=0.3,0.4,0.6,0.8$, and $0.9$ for sample sizes $n=2000,5000$, and 10,000 . Furthermore, it can be observed that, for $\alpha=1.25,1.5$, $1.75$ and 1.9, the RMSE of the modified truncated is less than that of the Hill estimator for different relocations for $\rho=0.6,0.7,0.8$, and $0.9$ for smaller sample size and even for $\rho=0.5$ for larger sample size. This clearly indicates the efficiency of the modified truncated estimator over the Hill estimator for higher values of the concentration parameter $\rho$.

It can be observed in Table 3 that the RMSE of the modified truncated estimator is less than that of the characteristic function-based estimator for almost all values of $\alpha$ corresponding to all values of $\sigma$.
The Hill estimator (Dufour and Kurz-Kim $(2010)$ ) is defined for $1 \leq \alpha \leq 2$, whereas the modified truncated estimator is defined for the whole range $0 \leq \alpha \leq 2$. In addition, the overall performance of the modified truncated estimator is quite good in terms of efficiency and consistency over both the Hill estimator and the characteristic function-based estimator.

Thus, we have established an estimator of the index parameter $\alpha$ that strongly supports its parameter space $(0,2]$. It can be observed from the above real life data applications that the modified truncated estimator is quite close to that of the characteristic function-based estimator. In addition, it is simpler and computationally easier than that of the estimator defined in Anderson and Arnold (1993). Thus, it may be considered as a better estimator.

Again, when the estimator of $a$ lies between 1 and 2 , is attempted to model a mixture of two distributions with the value of the index parameter as that of the two extreme tails that is modeling a mixture of Cauchy $(\alpha=1)$ and normal $(\alpha=2)$ distributions when $1<\alpha<2$ or modeling a mixture of Double Exponential $\left(\alpha=\frac{1}{2}\right)$ and Cauchy $(\alpha=1)$ distributions when $\frac{1}{2}<\alpha<1$. Then, it is compared with that of the stable family of distributions for goodness of fit.

We could have used the usual technique of non-linear optimization as used in Salimi et al. (2018) for estimation, but it is computationally demanding and also the (statistical) consistency of the estimators obtained by such method is unknown. In contrast, our proposed methods of trigonometric moment and modified truncated estimation are much simpler, computationally easier and also possess useful consistency properties and, even their asymptotic distributions can be presented in simple and elegant forms as already proved above.

统计代写|金融统计代写Financial Statistics代考|Background

In this section we provide briefly some background on Markov chains and results on stationarity of PHARCH models.
2.1. Markoo Chains
Suppose that $\mathbf{X}=\left{X_{n}, n \in \mathbb{Z}^{+}\right}, \mathbb{Z}^{+}:={0,1,2, \ldots}$ are random variables defined over $(\Omega, \mathcal{F}, \mathcal{B}(\Omega))$, and assume that $\mathrm{X}$ is a Markov chain with transition probability $P(x, A), x \in \Omega, A \subset \Omega$. Then we have the following definitions:

1. A function $f: \Omega \rightarrow \mathbb{R}$ is called the smallect semi-continuous function if $\liminf _{y \rightarrow x} f(y) \geq$ $f(x), x \in \Omega$. If $P(\cdot, A)$ is the smallest semi-continuous function for any open set $A \in \mathcal{B}(\Omega)$, we say that (the chain) $\mathbf{X}$ is a weak Feller chain.
2. A chain $\mathbf{X}$ is called $\varphi$-irreducible if there exists a measure $\varphi$ on $\mathcal{B}(\Omega)$ such that, for all $x$, whenever $\varphi(A)>0$, we have,
$$U(x, A)=\sum_{n=1}^{\infty} P^{n}(x, A)>0 .$$
3. The measure $\psi$ is called maximal with respect to $\varphi$, and we write $\psi>\varphi$, if $\psi(A)=0$ implies $\varphi(A)=0$, for all $A \in \mathcal{B}(\Omega)$. If $\mathbf{X}$ is $\varphi$-irreducible, then there exists a probability measure $\psi$, maximal, such that $\mathbf{X}$ is $\psi$-irreducible.
4. Let $d={d(n)}$ a distribution or a probability measure on $\mathbb{Z}^{+}$, and consider the Markov chain $\mathbf{X}{d}$ with transition kernel $$K{d}(x, A):=\sum_{n=0}^{\infty} P^{n}(x, A) d(n)$$
If there exits a transition kernel $T$ satisfying
$$K_{d}(x, A) \geq T(x, A), \quad x \in \Omega, A \in \mathcal{B}(\Omega),$$
then $T$ is called the continuous component of $K_{d}$.
5. If $\mathbf{X}$ is a Markov chain for which there exits a (sample) distribution $d$ such that $K_{d}$ has a continuous component $T$, with $T(x, \Omega)>0, \forall x$, then $\mathbf{X}$ is called a $T$-chain.
6. A measure $\pi$ over $\mathcal{B}(\Omega), \sigma$-finite, with the property
$$\pi(A)=\int_{\Omega} \pi(d x) P(x, A), A \in \mathcal{B}(\Omega)$$
is called an invariant measure.
The following two lemmas will be useful. See Meyn and Tweedie (1996) for the proofs and further details. We denote by $I_{A}(\cdot)$ the indicator function of $A$.

统计代写|金融统计代写Financial Statistics代考|Applications

8.1。黄金价格数据推断（以美元计）（1980-2013）

8.2. 白银价格数据推论（以美元计）（1980-2013）

统计代写|金融统计代写Financial Statistics代考|Findings and Concluding Remarks

Hill 估计器（Dufour 和 Kurz-Kim(2010)) 定义为1≤一个≤2，而修改的截断估计量是为整个范围定义的0≤一个≤2. 此外，改进的截断估计器的整体性能在效率和一致性方面都优于希尔估计器和基于特征函数的估计器。

统计代写|金融统计代写Financial Statistics代考|Background

2.1。Markoo 链

1. 一个函数F:Ω→R称为 smallect 半连续函数，如果林信息是→XF(是)≥ F(X),X∈Ω. 如果磷(⋅,一个)是任何开集的最小半连续函数一个∈乙(Ω)，我们说（链）X是一个弱 Feller 链。
2. 一条链子X叫做披-如果存在测度则不可约披上乙(Ω)这样，对于所有人X, 每当披(一个)>0， 我们有，
在(X,一个)=∑n=1∞磷n(X,一个)>0.
3. 的措施ψ被称为最大关于披，我们写ψ>披， 如果ψ(一个)=0暗示披(一个)=0， 对所有人一个∈乙(Ω). 如果X是披- 不可约，则存在概率测度ψ, 最大, 这样X是ψ- 不可约。
4. 让d=d(n)分布或概率测度从+，并考虑马尔可夫链Xd带有转换内核ķd(X,一个):=∑n=0∞磷n(X,一个)d(n)
如果存在过渡内核吨令人满意的
ķd(X,一个)≥吨(X,一个),X∈Ω,一个∈乙(Ω),
然后吨称为连续分量ķd.
5. 如果X是一个存在（样本）分布的马尔可夫链d这样ķd有一个连续的分量吨， 和吨(X,Ω)>0,∀X， 然后X被称为吨-链。
6. 一种方法圆周率超过乙(Ω),σ- 有限的，与财产
圆周率(一个)=∫Ω圆周率(dX)磷(X,一个),一个∈乙(Ω)
称为不变测度。
以下两个引理将很有用。有关证明和更多详细信息，请参见 Meyn 和 Tweedie (1996)。我们表示我一个(⋅)的指标函数一个.

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