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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|多元统计分析代写Multivariate Statistical Analysis代考|Hotelling’s T 2-Distribution

Suppose that $Y \in \mathbb{R}^p$ is a standard normal random vector, i.e. $Y \sim N_p(0, \mathcal{I})$, independent of the random matrix $\mathcal{M} \sim W_p(\mathcal{I}, n)$. What is the distribution of $Y^{\top} \mathcal{M}^{-1} Y$ ? The answer is provided by the Hotelling $T^2$-distribution: $n Y^{\top} \mathcal{M}^{-1} Y$ is Hotelling $T_{p, n}^2$ distributed.

The Hotelling $T^2$-distribution is a generalisation of the Student $t$-distribution. The general multinormal distribution $N(\mu, \Sigma)$ is considered in Theorem 5.8. The Hotelling $T^2$-distribution will play a central role in hypothesis testing in Chap. $7 .$
Theorem $5.8$ If $X \sim N_p(\mu, \Sigma)$ is independent of $\mathcal{M} \sim W_p(\Sigma, n)$, then
$$n(X-\mu)^{\top} \mathcal{M}^{-1}(X-\mu) \sim T_{p, n}^2$$
Corollary 5.3 If $\bar{x}$ is the mean of a sample drawn from a normal population $N_p(\mu, \Sigma)$ and $\mathcal{S}$ is the sample covariance matrix, then
$$(n-1)(\bar{x}-\mu)^{\top} \mathcal{S}^{-1}(\bar{x}-\mu)=n(\bar{x}-\mu)^{\top} \mathcal{S}u^{-1}(\bar{x}-\mu) \sim T{p, n-1}^2$$
Recall that $\mathcal{S}u=\frac{n}{n-1} \mathcal{S}$ is an unbiased estimator of the covariance matrix. A connection between the Hotelling $T^2$-and the $F$-distribution is given by the next theorem. Theorem $5.9$ $$T{p, n}^2=\frac{n p}{n-p+1} F_{p, n-p+1} .$$
Example 5.5 In the univariate case $(p=1)$, this theorem boils down to the well-known result:
$$\left(\frac{\bar{x}-\mu}{\sqrt{\mathcal{S}u} / \sqrt{n}}\right)^2 \sim T{1, n-1}^2=F_{1, n-1}=t_{n-1}^2$$
For further details on Hotelling $T^2$-distribution see Mardia et al. (1979). The next corollary follows immediately from (3.23), (3.24) and from Theorem 5.8. It will be useful for testing linearar restrictions in multinormál populations.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Spherical and Elliptical Distributions

The multinormal distribution belongs to the large family of elliptical distributions which has recently gained a lot of attention in financial mathematics. Elliptical distributions are often used, particularly in risk management.

Definition 5.1 A $(p \times 1)$ random vector $Y$ is said to have a spherical distribution $S_p(\phi)$ if its characteristic function $\psi_Y(t)$ satisfies: $\psi_Y(t)=\phi\left(t^{\top} t\right)$ for some scalar function $\phi(.)$ which is then called the characteristic generator of the spherical distribution $S_p(\phi)$. We will write $Y \sim S_p(\phi)$.

This is only one of several possible ways to define spherical distributions. We can see spherical distributions as an extension of the standard multinormal distribution $N_p\left(0, \mathcal{I}_p\right)$.
Theorem 5.10 Spherical random variables have the following properties:

1. All marginal distributions of a spherically distributed random vector are spherical.
2. All the marginal characteristic functions have the same generator:
3. Let $X \sim S_p(\phi)$, then $X$ has the same distribution as $r^{(p)}$ where $u^{(p)}$ is a random vector distributed uniformly on the unit sphere surface in $\mathbb{R}^p$ and $r \geq 0$ is a random variable independent of $u^{(p)}$. If $\mathrm{E}\left(r^2\right)<\infty$, then
$$\mathrm{E}(X)=0, \quad \operatorname{Cov}(X)=\frac{\mathrm{E}\left(r^2\right)}{p} \mathcal{I}_p .$$
The random radius $r$ is related to the generator $\phi$ by a relation described in Fang, Kotz, and $\mathrm{Ng}$ (1990, p. 29). The moments of $X \sim S_p(\phi)$, provided that they exist, can be expressed in terms of one-dimensional integral.

A spherically distributed random vector does not, in general, necessarily possess a density. However, if it does, the marginal densities of dimension smaller than $p-1$ are continuous and the marginal densities of dimension smaller than $p-2$ are differentiable (except possibly at the origin in both cases). Univariate marginal densities for $p$ greater than 2 are non-decreasing on $(-\infty, 0)$ and non-increasing on $(0, \infty)$

# 多元统计分析代考

## 统计代写|多元统计分析代写多元统计分析代考|Hotelling ‘s t2 -distribution

$$n(X-\mu)^{\top} \mathcal{M}^{-1}(X-\mu) \sim T_{p, n}^2$$

$$(n-1)(\bar{x}-\mu)^{\top} \mathcal{S}^{-1}(\bar{x}-\mu)=n(\bar{x}-\mu)^{\top} \mathcal{S}u^{-1}(\bar{x}-\mu) \sim T{p, n-1}^2$$

$$\left(\frac{\bar{x}-\mu}{\sqrt{\mathcal{S}u} / \sqrt{n}}\right)^2 \sim T{1, n-1}^2=F_{1, n-1}=t_{n-1}^2$$

## 统计代写|多元统计分析代写多元统计分析代考|球形和椭圆分布

$$\mathrm{E}(X)=0, \quad \operatorname{Cov}(X)=\frac{\mathrm{E}\left(r^2\right)}{p} \mathcal{I}_p .$$

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