 Hotelling's T-squared statistic

For independent multivariate normal observations with positive-definite covariance, this statistic is the maximum squared studentized mean contrast. With unbiased sample covariance matrix $S$ and $n>p$, its null distribution obeys $(n-p)T^2/(p(n-1))\sim F_{p,n-p}$. This follows from independence of the Gaussian sample mean and the residual Wishart distribution.