De-biasing the lasso with degrees-of-freedom adjustment

This paper studies schemes to de-bias the Lasso in sparse linear regression with Gaussian design where goal is estimate and construct confidence intervals for a low-dimensional projection of unknown coefficient vector preconceived direction a0. Our analysis reveals that previously analyzed propositions require modification order enjoy nominal coverage asymptotic efficiency full range level sparsity. takes form degrees-of-freedom adjustment accounts dimension model selected by Lasso. The (a) preserves success de-biasing methodologies regimes previous proposals were successful, (b) repairs provides produce spurious inferences provably fail achieve coverage. Hence our theoretical simulation results call implementation this methodologies. Let s0 denote number nonzero coefficients true ? population Gram matrix. unadjusted scheme may as soon s0?n2/3 if known. If unknown, grants contrast general a0 when s0logp n+min{s?logp n,???1 ?1 logp ? ??1/2 ?2 n}+min(s?,s0)logp n?0 s?=? ??1a0?0. dependence s0,s? ???1a0?1 optimal closes gap upper lower bounds. construction estimated score novel methodology handle dense directions Beyond adjustment, proof techniques yield sharp ?? error bound which independent interest.