A Bootstrap Lasso + Partial Ridge Method to Construct Confidence Intervals for Parameters in High-dimensional Sparse Linear Models

摘要

For high-dimensional sparse linear models, how to construct confidenceintervals for coefficients remains a difficult question. The main reason is thecomplicated limiting distributions of common estimators such as the Lasso.Several confidence interval construction methods have been developed, andBootstrap Lasso+OLS is notable for its simple technicality, goodinterpretability, and comparable performance with other more complicatedmethods. However, Bootstrap Lasso+OLS depends on the beta-min assumption, atheoretic criterion that is often violated in practice. In this paper, weintroduce a new method called Bootstrap Lasso+Partial Ridge (LPR) to relax thisassumption. LPR is a two-stage estimator: first using Lasso to select featuresand subsequently using Partial Ridge to refit the coefficients. Simulationresults show that Bootstrap LPR outperforms Bootstrap Lasso+OLS when thereexist small but non-zero coefficients, a common situation violating thebeta-min assumption. For such coefficients, compared to Bootstrap Lasso+OLS,confidence intervals constructed by Bootstrap LPR have on average 50% largercoverage probabilities. Bootstrap LPR also has on average 35% shorterconfidence interval lengths than the de-sparsified Lasso methods, regardless ofwhether linear models are misspecified. Additionally, we provide theoreticalguarantees of Bootstrap LPR under appropriate conditions and implement it inthe R package "HDCI."