  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.
