Sparsified simultaneous confidence intervals for high-dimensional linear models

Statistical inference of the high-dimensional regression coefficients is challenging because the uncertainty introduced by the model selection procedure is hard to account for. Currently, the inference of the model and the inference of the coefficients are separately sought. A critical question remains unsettled; that is, is it possible to embed the inference of the model into the simultaneous inference of the coefficients? If so, then how to properly design a simultaneous inference tool with desired properties? To this end, we propose a notion of simultaneous confidence intervals called the sparsified simultaneous confidence intervals (SSCI). Our intervals are sparse in the sense that some of the intervals' upper and lower bounds are shrunken to zero (i.e., [0, 0]), indicating the unimportance of the corresponding covariates. These covariates should be excluded from the final model. The rest of the intervals, either containing zero (e.g., [-1,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[-1,1]$$\end{document} or [0, 1]) or not containing zero (e.g., [2, 3]), indicate the plausible and significant covariates, respectively. The SSCI intuitively suggests a lower-bound model with significant covariates only and an upper-bound model with plausible and significant covariates. The proposed method can be coupled with various selection procedures, making it ideal for comparing their uncertainty. For the proposed method, we establish desirable asymptotic properties, develop intuitive graphical tools for visualization, and justify its superior performance through simulation and real data analysis.