Assuming only the existence and properties of the univariate normal distribution, define $\mathcal{N}_{p}(\underline{\mu}, \Sigma)$, the multivariate normal distribution with mean (row-)vector $\underline{\mu}$ and dispersion matrix $\Sigma$; and $W_{p}(\nu ; \Sigma)$, the Wishart distribution on integer $\nu>1$ degrees of freedom and with scale parameter $\Sigma$. Show that, if $\underline{X} \sim \mathcal{N}_{p}(\underline{\mu}, \Sigma), S \sim W_{p}(\nu ; \Sigma)$, and $\underline{b}(1 \times q), A(p \times q)$ are fixed, then $\underline{b}+\underline{X} A \sim \mathcal{N}_{q}(\underline{b}+\underline{\mu} A, \Phi), A^{\mathrm{T}} S A \sim W_{p}(\nu ; \Phi)$, where $\Phi=A^{\mathrm{T}} \Sigma A$.

The random $(n \times p)$ matrix $X$ has rows that are independently distributed as $\mathcal{N}_{p}(\underline{\mathrm{M}}, \Sigma)$, where both parameters $\underline{\mathrm{M}}$ and $\Sigma$ are unknown. Let $\underline{\bar{X}}:=n^{-1} \mathbf{1}^{\mathrm{T}} X$, where 1 is the $(n \times 1)$ vector of $1 \mathrm{~s}$; and $S^{c}:=X^{\mathrm{T}} \Pi X$, with $\Pi:=I_{n}-n^{-1} 11^{\mathrm{T}}$. State the joint distribution of $\bar{X}$ and $S^{c}$ given the parameters.

Now suppose $n>p$ and $\Sigma$ is positive definite. Hotelling's $T^{2}$ is defined as

$$T^{2}:=n(\underline{\bar{X}}-\underline{\mathrm{M}})\left(\bar{S}^{c}\right)^{-1}(\underline{\bar{X}}-\underline{\mathrm{M}})^{\mathrm{T}}$$

where $\bar{S}^{c}:=S^{c} / \nu$ with $\nu:=(n-1)$. Show that, for any values of $\underline{\mathrm{M}}$ and $\Sigma$,

$$\left(\frac{\nu-p+1}{\nu p}\right) T^{2} \sim F_{\nu-p+1}^{p},$$

the $F$ distribution on $p$ and $\nu-p+1$ degrees of freedom.

[You may assume that:

1. If $S \sim W_{p}(\nu ; \Sigma)$ and $\mathbf{a}$ is a fixed $(p \times 1)$ vector, then

$$\frac{\mathbf{a}^{\mathrm{T}} \Sigma^{-1} \mathbf{a}}{\mathbf{a}^{\mathrm{T}} S^{-1} \mathbf{a}} \sim \chi_{\nu-p+1}^{2}$$

2. If $V \sim \chi_{p}^{2}, W \sim \chi_{\lambda}^{2}$ are independent, then

$$\frac{V / p}{W / \lambda} \sim F_{\lambda}^{p} \text {. }$$