# 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|MAST90085

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Student’s t-distribution

The $t$-distribution was first analyzed by Gosset (1908) who published it under pseudonym “Student” by request of his employer. Let $X$ be a normally distributed random variable with mean $\mu$ and variance $\sigma^2$, and $Y$ be the random variable such that $Y^2 / \sigma^2$ has a chi-square distribution with $n$ degrees of freedom. Assume that $X$ and $Y$ are independent, then
$$t \stackrel{\text { def }}{=} \frac{X \sqrt{n}}{Y}$$
is distributed as Student’s $t$ with $n$ degrees of freedom. The $t$-distribution has the following density function :
$$f_t(x ; n)=\frac{\Gamma\left(\frac{n+1}{2}\right)}{\sqrt{n \pi} \Gamma\left(\frac{n}{2}\right)}\left(1+\frac{x^2}{n}\right)^{-\frac{n+1}{2}}$$
where $n$ is the number of degrees of freedom, $-\infty4)$ are
\begin{aligned} \mu & =0 \ \sigma^2 & =\frac{n}{n-2} \ \text { Skewness } & =0 \ \text { Kurtosis } & =3+\frac{6}{n-4} \end{aligned}
The $t$-distribution is symmetric around 0 , which is consistent with the fact that its mean is 0 and skewness is also 0 (Fig. 4.8).

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

The cumulative distribution function (cdf) of a two-dimensional vector $\left(X_1, X_2\right)$ is given by
$$F\left(x_1, x_2\right)=\mathrm{P}\left(X_1 \leq x_1, X_2 \leq x_2\right) .$$
For the case that $X_1$ and $X_2$ are independent, their joint cumulative distribution function $F\left(x_1, x_2\right)$ can be written as a product of their 1-dimensional marginals:
$$F\left(x_1, x_2\right)=F_{X_1}\left(x_1\right) F_{X_2}\left(x_2\right)=\mathrm{P}\left(X_1 \leq x_1\right) \mathrm{P}\left(X_2 \leq x_2\right) .$$
But how can we model dependence of $X_1$ and $X_2$ ? Most people would suggest linear correlation. Correlation is though an appropriate measure of dependence only when the random variables have an elliptical or spherical distribution, which include the normal multivariate distribution. Although the terms “correlation” and “dependency” are often used interchangeably, correlation is actually a rather imperfect measure of dependency, and there are many circumstances where correlation should not be used. Copulae represent an elegant concept of connecting marginals with joint cumulative distribution functions. Copulae are functions that join or “couple” multivariate distribution functions to their 1-dimensional marginal distribution functions. Let us consider a $d$-dimensional vector $X=\left(X_1, \ldots, X_d\right)^{\top}$. Using copulae, the marginal distribution functions $F_{X_i}(i=1, \ldots, d)$ can be separately modeled from their dependence structure and then coupled together to form the multivariate distribution $F_X$. Copula functions have a long history in probability theory and statistics. Their application in finance is very recent. Copulae are important in Value-at-Risk calculations and constitute an essential tool in quantitative finance (Härdle et al. 2009).

First let us concentrate on the two-dimensional case, then we will extend this concept to the $d$-dimensional case, for a random variable in $\mathbb{R}^d$ with $d \geq 1$. To be able to define a copula function, first we need to represent a concept of the volume of a rectangle, a 2-increasing function and $a$ grounded function.

Let $U_1$ and $U_2$ be two sets in $\overline{\mathbb{R}}=\mathbb{R} \cup{+\infty} \cup{-\infty}$ and consider the function $F: U_1 \times U_2 \longrightarrow \overline{\mathbb{R}}$

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Student’s t-distribution

$$t \stackrel{\text { def }}{=} \frac{X \sqrt{n}}{Y}$$

$$f_t(x ; n)=\frac{\Gamma\left(\frac{n+1}{2}\right)}{\sqrt{n \pi} \Gamma\left(\frac{n}{2}\right)}\left(1+\frac{x^2}{n}\right)^{-\frac{n+1}{2}}$$

$$\mu=0 \sigma^2 \quad=\frac{n}{n-2} \text { Skewness }=0 \text { Kurtosis } \quad=3+\frac{6}{n-4}$$

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

$$F\left(x_1, x_2\right)=\mathrm{P}\left(X_1 \leq x_1, X_2 \leq x_2\right) .$$

$$F\left(x_1, x_2\right)=F_{X_1}\left(x_1\right) F_{X_2}\left(x_2\right)=\mathrm{P}\left(X_1 \leq x_1\right) \mathrm{P}\left(X_2 \leq x_2\right) .$$

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