  For a high-dimensional parameter of interest, tests based on quadratic
statistics are known to have low power against subsets of the parameter space
(henceforth, parameter subspaces). In addition, they typically involve an
inverse covariance matrix which is difficult to estimate in high-dimensional
settings. I simultaneously address these two issues by proposing a novel test
statistic that is large in a conic parameter subspace of interest. This test
statistic generalizes the Wald statistic and nests many well-known test
statistics. For a given parameter subspace, the statistic is free of tuning
parameters and suitable for high-dimensional settings if the subspace is
sufficiently small. It can be computed using regularized linear regression,
where the type of regularization and the regularization parameters are
completely determined by the parameter subspace of interest. I illustrate the
statistic on subspaces that consist of sparse or nearly-sparse vectors, for
which the computation corresponds to $\ell_0$- and $\ell_1$-regularized
regression, respectively.
