Papers › The ℓ-test: leveraging sparsity in the Gaussian linear model for improved inference
The ℓ-test: leveraging sparsity in the Gaussian linear model for improved inference
Souhardya Sengupta, Lucas Janson
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We develop novel LASSO-based methods for coefficient testing and confidence interval construction in the Gaussian linear model with n≥d. Our methods' finite-sample validity is identical to that of their ubiquitous ordinary-least-squares-t-test-based analogues, yet have substantially higher power when the true coefficient vector is sparse. In particular, under sparsity our coefficient test, which we call the ℓ-test, performs like the \emph{one-sided} t-test (despite not being given any information about the sign), and ℓ-test-based confidence intervals are correspondingly shorter than the standard t-test-based intervals. The nature of the ℓ-test directly provides a novel exact adjustment conditional on LASSO selection for post-selection inference, allowing for the construction of post-selection p-values and confidence intervals. None of our methods require resampling or Monte Carlo estimation. We perform a variety of simulations and a real data analysis on an HIV drug resistance data set to demonstrate the benefits of the ℓ-test. We additionally show that the ℓ-test can be applied to a large class of asymptotically Gaussian estimators, dramatically expanding its applicability beyond linear models.
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