# 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|STAT4102

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|The Wishart Distribution

The Wishart distribution (named after its discoverer) plays a prominent role in the analysis of the estimated covariance matrices. If the mean of $X \sim N_p(\mu, \Sigma)$ is known to be $\mu=0$, then for a data matrix $\mathcal{X}(n \times p)$ the estimated covariance matrix is proportional to $\mathcal{X}^{\top} \mathcal{X}$. This is the point where the Wishart distribution comes in, because $\mathcal{M}(p \times p)=\mathcal{X}^{\top} \mathcal{X}=\sum_{i=1}^n x_i x_i^{\top}$ has a Wishart distribution $W_p(\Sigma, n)$.
Example 5.4 Set $p=1$, then for $X \sim N_1\left(0, \sigma^2\right)$ the data matrix of the observations
$$\mathcal{X}=\left(x_1, \ldots, x_n\right)^{\top} \quad \text { with } \mathcal{M}=\mathcal{X}^{\top} \mathcal{X}=\sum_{i=1}^n x_i x_i$$
leads to the Wishart distribution $W_1\left(\sigma^2, n\right)=\sigma^2 \chi_n^2$. The one-dimensional Wishart distribution is thus in fact a $\chi^2$ distribution.

When we talk about the distribution of a matrix, we mean of course the joint distribution of all its elements. More exactly: since $\mathcal{M}=\mathcal{X}^{\top} \mathcal{X}$ is symmetric we only need to consider the elements of the lower triangular matrix

Hence the Wishart distribution is defined by the distribution of the vector
$$\left(m_{11}, \ldots, m_{p 1}, m_{22}, \ldots, m_{p 2}, \ldots, m_{p p}\right)^{\top} .$$
Linear transformations of the data matrix $\mathcal{X}$ also lead to Wishart matrices.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Hotelling’s T2-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 generalization 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 $5.9$ $$T{p, n}^2=\frac{n p}{n-p+1} F_{p, n-p+1} .$$

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|The Wishart Distribution

Wishart 分布 (以其发现者的名字命名) 在估计协方差矩阵的分析中起着重要作用。如果均值 $X \sim N_p(\mu, \Sigma)$ 众所周知 $\mu=0$, 那么对于一个数据矩阵 $\mathcal{X}(n \times p)$ 估计的协方差矩阵与 $\mathcal{X}^{\top} \mathcal{X}$. 这就 是 Wishart 分布的用武之地，因为 $\mathcal{M}(p \times p)=\mathcal{X}^{\top} \mathcal{X}=\sum_{i=1}^n x_i x_i^{\top}$ 服从 Wishart 分布 $W_p(\Sigma, n)$

$$\mathcal{X}=\left(x_1, \ldots, x_n\right)^{\top} \quad \text { with } \mathcal{M}=\mathcal{X}^{\top} \mathcal{X}=\sum_{i=1}^n x_i x_i$$

$$\left(m_{11}, \ldots, m_{p 1}, m_{22}, \ldots, m_{p 2}, \ldots, m_{p p}\right)^{\top} .$$

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Hotelling’s T2-Distribution

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

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

$$T p, n^2=\frac{n p}{n-p+1} F_{p, n-p+1} .$$

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