Minimax Character of Hotelling's $T^2$ Test in the Simplest Case

Abstract

In the first nontrivial case, dimension $p = 2$ and sample size $N = 3$, it is proved that Hotelling's $T^2$ test of level $\alpha$ maximizes, among all level $\alpha$ tests, the minimum power on each of the usual contours where the $T^2$ test has constant power. A corollary is that the $T^2$ test is most stringent of level $\alpha$ in this case.