On inadmissibility of Hotelling T2-tests for restricted alternatives

For multinormal distributions, testing against a global shift alternative, the Hotelling T2-test is uniformly most powerful invariant, and hence admissible. For testing against restricted alternatives this feature may no longer be true. It is shown that whenever the dispersion matrix is an M-matrix, Hotellingʹs T2-test is inadmissible, though some union–intersection tests may not be so.