A Flexible Framework for Hypothesis Testing in High-dimensions
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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 ℓ_∞ 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 ( p)/√(n)→ 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.
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