On High Dimensional Post-Regularization Prediction Intervals

This paper considers the construction of prediction intervals for future observations in high dimensional regression models. We propose a new approach to evaluate the uncertainty for estimating the mean parameter based on the widely-used penalization/regularization methods. The proposed method is then applied to construct prediction intervals for sparse linear models as well as sparse additive models. We establish the asymptotic normality of the estimator for the mean parameter and the asymptotic coverage probability of the prediction intervals. The theoretical properties of the proposed methods are verified by extensive simulation studies and real data analysis. Keyword: Asymptotic normality, Prediction, Lasso, SCAD, Dantzig selector, Sparse additive model, Spline, Linear model