Title: A Bootstrap Lasso + Partial Ridge Method to Construct Confidence Intervals for Parameters in High-dimensional Sparse Linear Models,
Abstract: For high-dimensional sparse linear models, how to construct confidence
intervals for coefficients remains a difficult question. The main reason is the
complicated limiting distributions of common estimators such as the Lasso.
Several confidence interval construction methods have been developed, and
Bootstrap Lasso+OLS is notable for its simple technicality, good
interpretability, and comparable performance with other more complicated
methods. However, Bootstrap Lasso+OLS depends on the beta-min assumption, a
theoretic criterion that is often violated in practice. In this paper, we
introduce a new method called Bootstrap Lasso+Partial Ridge (LPR) to relax this
assumption. LPR is a two-stage estimator: first using Lasso to select features
and subsequently using Partial Ridge to refit the coefficients. Simulation
results show that Bootstrap LPR outperforms Bootstrap Lasso+OLS when there
exist small but non-zero coefficients, a common situation violating the
beta-min assumption. For such coefficients, compared to Bootstrap Lasso+OLS,
confidence intervals constructed by Bootstrap LPR have on average 50% larger
coverage probabilities. Bootstrap LPR also has on average 35% shorter
confidence interval lengths than the de-sparsified Lasso methods, regardless of
whether linear models are misspecified. Additionally, we provide theoretical
guarantees of Bootstrap LPR under appropriate conditions and implement it in
the R package "HDCI."