The $L$-test: Increasing the Linear Model $F$-test's Power Under Sparsity Without Sacrificing Validity

We introduce a new procedure for testing the significance of a set of regression coefficients in a Gaussian linear model with $n \geq d$. Our method, the $L$-test, provides the same statistical validity guarantee as the classical $F$-test, while attaining higher power when the nuisance coefficients are sparse. Although the $L$-test requires Monte Carlo sampling, each sample's runtime is dominated by simple matrix-vector multiplications so that the overall test remains computationally efficient. Furthermore, we provide a Monte-Carlo-free variant that can be used for particularly large-scale multiple testing applications. We give intuition for the power of our approach, validate its advantages through extensive simulations, and illustrate its practical utility in both single- and multiple-testing contexts with an application to an HIV drug resistance dataset. In the concluding remarks, we also discuss how our methodology can be applied to a more general class of parametric models that admit asymptotically Gaussian estimators.