Title: Statistical inference for high dimensional regression via Constrained Lasso,
Abstract: In this paper, we propose a new method for estimation and constructing
confidence intervals for low-dimensional components in a high-dimensional
model. The proposed estimator, called Constrained Lasso (CLasso) estimator, is
obtained by simultaneously solving two estimating equations---one imposing a
zero-bias constraint for the low-dimensional parameter and the other forming an
$\ell_1$-penalized procedure for the high-dimensional nuisance parameter. By
carefully choosing the zero-bias constraint, the resulting estimator of the low
dimensional parameter is shown to admit an asymptotically normal limit
attaining the Cramér-Rao lower bound in a semiparametric sense. We propose
a tuning-free iterative algorithm for implementing the CLasso. We show that
when the algorithm is initialized at the Lasso estimator, the de-sparsified
estimator proposed in van de Geer et al. [\emph{Ann. Statist.} {\bf 42} (2014)
1166--1202] is asymptotically equivalent to the first iterate of the algorithm.
We analyse the asymptotic properties of the CLasso estimator and show the
globally linear convergence of the algorithm. We also demonstrate encouraging
empirical performance of the CLasso through numerical studies.