On the feasibility of parsimonious variable selection for Hotelling's $T^2$-test

Hotelling's $T^2$-test for the mean of a multivariate normal distribution is one of the triumphs of classical multivariate analysis. It is uniformly most powerful among invariant tests, and admissible, proper Bayes, and locally and asymptotically minimax among all tests. Nonetheless, investigators often prefer non-invariant tests, especially those obtained by selecting only a small subset of variables from which the $T^2$-statistic is to be calculated, because such reduced statistics are more easily interpretable for their specific application. Thus it is relevant to ask the extent to which power is lost when variable selection is limited to very small subsets of variables, e.g. of size one (yielding univariate Student-$t^2$ tests) or size two (yielding bivariate $T^2$-tests). This study presents some evidence, admittedly fragmentary and incomplete, suggesting that in some cases no power may be lost over a wide range of alternatives.