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

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## 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 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
$$\mathcal{X}=\left(x_1, \ldots, x_n\right)^{\top} \quad \text { with } \quad \mathcal{M}=\mathcal{X}^{\top} \mathcal{X}=\sum_{i=1}^n x_i x_i$$
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
$$\mathcal{M}=\left(\begin{array}{cccc} m_{11} & & & \ m_{21} & m_{22} & & \ \vdots & \vdots & \ddots & \ m_{p 1} & m_{p 2} & \ldots & m_{p p} \end{array}\right)$$
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 $T^2(p, n)$ 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 Chapter 7.
THEOREM 5.8 If $X \sim N_p(\mu, \Sigma)$ is independent of $\mathcal{M} \sim W_p(\Sigma, n)$, then
$$n(X-\mu)^{\top} \mathcal{M}^{-1}(X-\mu) \sim T^2(p, n)$$
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
$$(n-1)(\bar{x}-\mu)^{\top} \mathcal{S}^{-1}(\bar{x}-\mu)=n(\bar{x}-\mu)^{\top} \mathcal{S}u^{-1}(\bar{x}-\mu) \sim T^2(p, n-1) .$$ 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 $$T^2(p, n)=\frac{n p}{n-p+1} F{p, n-p+1}$$
EXAMPLE 5.5 In the univariate case $(p=1)$, this theorem boils down to the well known result:
$$\left(\frac{\bar{x}-\mu}{\sqrt{\mathcal{S}u} / \sqrt{n}}\right)^2 \sim T^2(1, n-1)=F{1, n-1}=t_{n-1}^2$$
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 linear restrictions in multinormal populations.

# 多元统计分析代考

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

Wishart分布(以其发现者命名)在估计协方差矩阵的分析中起着重要作用。如果已知$X \sim N_p(\mu, \Sigma)$的平均值为$\mu=0$，则对于数据矩阵$\mathcal{X}(n \times p)$，估计的协方差矩阵与$\mathcal{X}^{\top} \mathcal{X}$成正比。这就是Wishart分布的切入点，因为$\mathcal{M}(p \times p)=\mathcal{X}^{\top} \mathcal{X}=\sum_{i=1}^n x_i x_i^{\top}$有一个Wishart分布$W_p(\Sigma, n)$。

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

$$\mathcal{M}=\left(\begin{array}{cccc} m_{11} & & & \ m_{21} & m_{22} & & \ \vdots & \vdots & \ddots & \ m_{p 1} & m_{p 2} & \ldots & m_{p p} \end{array}\right)$$

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

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

Hotelling $T^2$ -分布是Student $t$ -分布的一般化。定理5.8中考虑了一般多正态分布$N(\mu, \Sigma)$。Hotelling $T^2$分布将在第7章的假设检验中发挥核心作用。

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

$$(n-1)(\bar{x}-\mu)^{\top} \mathcal{S}^{-1}(\bar{x}-\mu)=n(\bar{x}-\mu)^{\top} \mathcal{S}u^{-1}(\bar{x}-\mu) \sim T^2(p, n-1) .$$回想一下$\mathcal{S}u=\frac{n}{n-1} \mathcal{S}$是协方差矩阵的无偏估计量。下一个定理给出了Hotelling $T^2$ -和$F$ -分布之间的联系。定理5.9 $$T^2(p, n)=\frac{n p}{n-p+1} F{p, n-p+1}$$ 在单变量情况下$(p=1)$，这个定理可以归结为众所周知的结果: $$\left(\frac{\bar{x}-\mu}{\sqrt{\mathcal{S}u} / \sqrt{n}}\right)^2 \sim T^2(1, n-1)=F{1, n-1}=t{n-1}^2$$

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