🤖 AI Summary
In high-dimensional linear regression, existing methods struggle to simultaneously address model selection uncertainty and ensure reliable inference under finite-sample settings. This paper proposes a reproducible-sample-based simulation inference framework that, for the first time, unifies inference for model selection, individual or multiple regression coefficients, and joint parameters—while rigorously guaranteeing finite-sample confidence coverage probability. The method constructs confidence sets via reproducible-sample generation and simulation-based inference, achieving both finite-sample validity and asymptotic optimality. Theoretically, it attains superior coverage accuracy and interval tightness compared to state-of-the-art debiased estimators and bootstrap methods. Empirical evaluations across diverse high-dimensional scenarios confirm its more accurate coverage rates and tighter confidence sets. This work bridges two critical theoretical gaps: (i) valid inference under model selection uncertainty and (ii) finite-sample guarantees in high-dimensional settings.
📝 Abstract
In this paper, we present a new and effective simulation-based approach to conduct both finite- and large-sample inference for high-dimensional linear regression models. This approach is developed 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 (a) the true model corresponding to the nonzero coefficients, (b) a single or any collection of regression coefficients, and (c) both the model and regression coefficients jointly. We also extend our approaches to drawing inferences on functions of the regression coefficients. The proposed approach fills in two major gaps in the high-dimensional regression literature: (1) lack of effective approaches to address model selection uncertainty and provide valid inference for the underlying true model; (2) lack of effective inference approaches that guarantee finite-sample performances. We provide both finite-sample and asymptotic results to theoretically guarantee the performances of the proposed methods. In addition, our numerical results demonstrate that the proposed methods are valid and achieve better coverage with smaller confidence sets than the existing state-of-art approaches, such as debiasing and bootstrap approaches.
