Hotelling's T^2

In multivariate analysis, similar role with T-statistic
Suppose $X \sim N_p(\mu,\Sigma),S \sim W_p(r,\Sigma) \text{ and } X \perp\!\!\!\perp S$

• pdf

  $T^2_{p,r}=(X-\mu)^T(S/r)^{-1}(X-\mu)$

• Properties

  ① When $p=1, T^2_{1,r}=\frac{N^2(0,1)}{\chi^2_r/r} \sim F_{1,r} \overset{d}{\equiv} t^2_r$
  ② $\frac{r-p+1}{p} \frac{T^2_{p,r}}{r} \sim F_{p,r-p+1} \text{ if } r>p-1$
  $\because$
  

• Application to multivariate normal sampling

  Suppose $X_1,\cdots,X_n \overset{iid}{\sim} N_p(\mu,\Sigma)$ where $n>p,\mu \in \mathbb{R}^p,\Sigma \in \mathbb{R}^{p \times p}:\text{positive definite},$ and $S=\frac{1}{n-1}\sum_{i=1}^{n}(X_i-\overline{X})(X_i-\overline{X})^T$
  
  $\frac{n-p}{n-1} \frac{T^2}{p} \sim F_{p,n-p}$ where $T^2=n(\overline{X}-\mu)^T S^{-1} (\overline{X}-\mu)$
  $\because$
  $\sqrt{n}(\overline{X}-\mu) \sim N_p(0,\Sigma), (n-1)S \sim W_p(n-1,\Sigma),\overline{X} \perp\!\!\!\perp S$
  => $\sqrt{n}(\overline{X}-\mu)^T((n-1)S/n-1)^{-1}\sqrt{n}(\overline{X}-\mu)=n(\overline{X}-\mu)^T S^{-1} (\overline{X}-\mu)$
  
  $P(n(\overline{X}-\mu)^T S^{-1} (\overline{X}-\mu) \leq \frac{(n-1)p}{n-p}F_{p,n-p}(\alpha))=1-\alpha$