Directing Power Towards Sub-Regions of the Alternative Hypothesis

In this paper, I propose a novel test statistic for testing hypotheses about a potentially high-dimensional parameter vector. To obtain 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-region of the alternative hypothesis (henceforth sub-alternative). Its existence depends on a restricted eigenvalue condition that is tied directly to the scope of the sub-alternative. I show that if the conventional sample covariance matrix is used, then the computation of the test statistic reduces to linear regression with a constant dependent variable, restricted by the same constraints that specify the sub-alternative. To demonstrate the test statistic, I consider the case of testing against a sparse sub-alternative, which is defined by the number of element-wise violations of a null hypothesis. In this case the computation reduces to $\ell_0$-regularized regression (best subset selection). In addition, I show that $\ell_1$-regularized regression (Lasso) corresponds to near-sparse sub-alternatives.