Directing Power Towards Sub-Alternatives

This paper proposes a novel test statistic for testing a potentially high-dimensional parameter vector. To derive the statistic, I generalize the Mahalanobis distance to measure length in a direction of interest. The test statistic is the sample analogue of the distance and directs power towards a sub-regions within the alternative hypothesis (sub-alternative). I show how the computation of this test statistic can reduce to a linear regression problem with a constant response vector, restricted by the same constraints that specify the sub-alternative. The existence of the statistic is directly tied to the scope of the sub-alternative and reduces to the Hotelling $T^2$ statistic if the sub-alternative coincides with the alternative. I demonstrate this test statistic by testing against sparse alternatives, where the computation reduces to $\ell_0$-regularized regression.