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Hotelling's $T^2$-test for the mean of a multivariate normal distribution isone of the triumphs of classical multivariate analysis. It is uniformly mostpowerful among invariant tests, and admissible, proper Bayes, and locally andasymptotically minimax among all tests. Nonetheless, investigators often prefernon-invariant tests, especially those obtained by selecting only a small subsetof variables from which the $T^2$-statistic is to be calculated, because suchreduced statistics are more easily interpretable for their specificapplication. Thus it is relevant to ask the extent to which power is lost whenvariable selection is limited to very small subsets of variables, e.g. of sizeone (yielding univariate Student-$t^2$ tests) or size two (yielding bivariate$T^2$-tests). This study presents some evidence, admittedly fragmentary andincomplete, suggesting that in some cases no power may be lost over a widerange of alternatives.

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