Step-down testing procedures for the Hotelling $T^2$ problem are considered. It is shown that the classical step-down procedure proposed by S. N. Roy, R. E. Bargmann and J. Roy is inadmissible among invariant procedures (for a suitable invariance group) in many situations, including all those for which more than two steps are contemplated. It is also noted that in most cases, the power of the step-down procedure decreases in at least one of the noncentrality parameters over part of the parameter space. Finally, several alternative admissible step-down procedures are proposed.
"On the Inadmissibility of Step-Down Procedures for the Hotelling $T^2$ Problem." Ann. Statist. 18 (1) 172 - 190, March, 1990. https://doi.org/10.1214/aos/1176347497