The \ell-test: Leveraging sparsity in the Gaussian linear model for improved inference

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 guarantees are identical to those of their ubiquitous ordinary least squares t test-based analogues, yet have substantially higher power when the true coefficient vector is sparse. In particular, our coefficient test, which we call the ℓ-test, performs like the one-sided t-test (despite not being given any information about the sign) under sparsity, and the corresponding confidence intervals are more than 10% 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.

This is joint work with Souhardya Sengupta.