### 统计代写|应用时间序列分析代写applied time series analysis代考|Scale Mixture of Normal Distributions

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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|应用时间序列分析代写applied time series anakysis代考|Scale Mixture of Normal Distributions

Recent studies of stock returns tend to use scale mixture or finite mixture of normal distributions. Under the assumption of scale mixture of normal distributions, the log return $r_{t}$ is normally distributed with mean $\mu$ and variance $\sigma^{2}\left[\right.$ i.e., $\left.r_{t} \sim N\left(\mu, \sigma^{2}\right)\right]$. However, $\sigma^{2}$ is a random variable that follows a positive distribution (e.g., $\sigma^{-2}$ follows a Gamma distribution). An example of finite mixture of normal distributions is
$$r_{t} \sim(1-X) N\left(\mu, \sigma_{1}^{2}\right)+X N\left(\mu, \sigma_{2}^{2}\right)$$
where $0 \leq \alpha \leq 1, \sigma_{1}^{2}$ is small and $\sigma_{2}^{2}$ is relatively large. For instance, with $\alpha=$ $0.05$, the finite mixture says that $95 \%$ of the returns follow $N\left(\mu, \sigma_{1}^{2}\right)$ and $5 \%$ follow $N\left(\mu, \sigma_{2}^{2}\right)$. The large value of $\sigma_{2}^{2}$ enables the mixture to put more mass at the tails of its distribution. The low percentage of returns that are from $N\left(\mu, \sigma_{2}^{2}\right)$ says that the majority of the returns follow a simple normal distribution. Advantages of mixtures of normal include that they maintain the tractability of normal, have finite higher order moments, and can capture the excess kurtosis. Yet it is hard to estimate the mixture parameters (e.g., the $\alpha$ in the finite-mixture case).

Figure $1.1$ shows the probability density functions of a finite mixture of normal, Cauchy, and standard normal random variable. The finite mixture of normal is $0.95 N(0,1)+0.05 N(0,16)$ and the density function of Cauchy is
$$f(x)=\frac{1}{\pi\left(1+x^{2}\right)}, \quad-\infty<x<\infty$$
It is seen that Cauchy distribution has fatter tails than the finite mixture of normal, which in turn has fatter tails than the standard normal.

## 统计代写|应用时间序列分析代写applied time series anakysis代考| Stable Distribution

The stable distributions are a natural generalization of normal in that they are stable under addition, which meets the need of continuously compounded returns $r_{t}$. Furthermore, stable distributions are capable of capturing excess kurtosis shown by historical stock returns. However, non-normal stable distributions do not have a finite variance, which is in conflict with most finance theories. In addition, statistical modeling using non-normal stable distributions is difficult. An example of non-normal stable distributions is the Cauchy distribution, which is symmetric with respect to its median, but has infinite variance.

## 统计代写|应用时间序列分析代写applied time series anakysis代考|Multivariate Returns

Let $\boldsymbol{r}{t}=\left(r{1 t}, \ldots, r_{N t}\right)^{\prime}$ be the log returns of $N$ assets at time $t$. The multivariate analyses of Chapters 8 and 9 are concerned with the joint distribution of $\left{\boldsymbol{r}{t}\right}{t=1}^{T}$. This joint distribution can be partitioned in the same way as that of Eq. (1.15). The analysis is then focused on the specification of the conditional distribution function $F\left(\boldsymbol{r}{t} \mid \boldsymbol{r}{t-1}, \ldots, \boldsymbol{r}{1}, \boldsymbol{\theta}\right)$. In particular, how the conditional expectation and conditional covariance matrix of $\boldsymbol{r}{t}$ evolve over time constitute the main subjects of Chapters 8 and $9 .$

The mean vector and covariance matrix of a random vector $X=\left(X_{1}, \ldots, X_{p}\right)$ are defined as
\begin{aligned} E(\boldsymbol{X}) &=\boldsymbol{\mu}{x}=\left[E\left(X{1}\right), \ldots, E\left(X_{p}\right)\right]^{\prime} \ \operatorname{Cov}(\boldsymbol{X}) &=\boldsymbol{\Sigma}{x}=E\left[\left(\boldsymbol{X}-\boldsymbol{\mu}{x}\right)\left(\boldsymbol{X}-\boldsymbol{\mu}{x}\right)^{\prime}\right] \end{aligned} provided that the expectations involved exist. When the data $\left{x{1}, \ldots, x_{T}\right}$ of $X$ are available, the sample mean and covariance matrix are defined as
$$\widehat{\boldsymbol{\mu}}{x}=\frac{1}{T} \sum{t=1}^{T} \boldsymbol{x}{t}, \quad \widehat{\boldsymbol{\Sigma}}{x}=\frac{1}{T} \sum_{t=1}^{T}\left(\boldsymbol{x}{t}-\widehat{\boldsymbol{\mu}}{x}\right)\left(\boldsymbol{x}{t}-\widehat{\boldsymbol{\mu}}{x}\right)^{\prime}$$These sample statistics are consistent estimates of their theoretical counterparts provided that the covariance matrix of $X$ exists. In the finance literature, multivariate normal distribution is often used for the log return $\boldsymbol{r}_{t}$.

## 统计代写|应用时间序列分析代写applied time series anakysis代考|Scale Mixture of Normal Distributions

r吨∼(1−X)ñ(μ,σ12)+Xñ(μ,σ22)

F(X)=1圆周率(1+X2),−∞<X<∞

## 统计代写|应用时间序列分析代写applied time series anakysis代考|Multivariate Returns

μ^X=1吨∑吨=1吨X吨,Σ^X=1吨∑吨=1吨(X吨−μ^X)(X吨−μ^X)′这些样本统计量是对其理论对应物的一致估计，前提是协方差矩阵X存在。在金融文献中，多元正态分布常用于对数回报r吨.

## 广义线性模型代考

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## MATLAB代写

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