Finite- and Large- Sample Inference for Model and Coefficients in High-dimensional Linear Regression with Repro Samples

This paper presents a new and effective simulation-based approach to conducting both finite- and large- sample inference for high-dimensional linear regression models. We develop this approach under the so-called repro samples framework, in which we conduct statistical inference by creating and studying the behavior of artificial samples that are obtained by mimicking the sampling mechanism of the data. We obtain confidence sets for either the true model, a single, or any collection of regression coefficients. The proposed approach addresses two major gaps in the high-dimensional regression literature: (1) lack of inference approaches that guarantee finite-sample performance; (2) lack of effective approaches to address model selection uncertainty and provide inference for the underlying true model. We provide both finite-sample and asymptotic results to theoretically guarantee the performance of the proposed methods. Besides enjoying theoretical advantages, our numerical results demonstrate that the proposed methods achieve better coverage with smaller confidence sets than the existing state-of-art approaches, such as debiasing and bootstrap approaches. We also extend our approaches to drawing inferences on functions of the regression coefficients.