Confidence Intervals for Sparse Penalized Regression With Random Designs

With the abundance of large data, sparse penalized regression techniques are commonly used in data analysis due to the advantage of simultaneous variable selection and estimation. A number of convex as well as nonconvex penalties have been proposed in the literature to achieve sparse estimates. Despite intense work in this area, how to perform valid inference for sparse penalized regression with a general penalty remains to be an active research problem. In this article, by making use of state-of-the-art optimization tools in stochastic variational inequality theory, we propose a unified framework to construct confidence intervals for sparse penalized regression with a wide range of penalties, including convex and nonconvex penalties. We study the inference for parameters under the population version of the penalized regression as well as parameters of the underlying linear model. Theoretical convergence properties of the proposed method are obtained. Several simulated and real data examples are presented to demonstrate the validity and effectiveness of the proposed inference procedure. Supplementary materials for this article are available online.