A Flexible Framework for Hypothesis Testing in High-dimensions

Hypothesis testing in the linear regression model is a fundamental statistical problem. We consider linear regression in the high-dimensional regime where the number of parameters exceeds the number of samples ($p> n$) and assume that the high-dimensional parameters vector is $s_0$ sparse. We develop a general and flexible $\ell_\infty$ projection statistic for hypothesis testing in this model. Our framework encompasses testing whether the parameter lies in a convex cone, testing the signal strength, testing arbitrary functionals of the parameter, and testing adaptive hypothesis. We show that the proposed procedure controls the type I error under the standard assumption of $s_0 (\log p)/\sqrt{n}\to 0$, and also analyze the power of the procedure. Our numerical experiments confirms our theoretical findings and demonstrate that we control false positive rate (type I error) near the nominal level, and have high power.