Drawing inferences for high‐dimensional linear models: A selection‐assisted partial regression and smoothing approach

Abstract

Drawing inferences for high‐dimensional models is challenging as regular asymptotic theories are not applicable. This article proposes a new framework of simultaneous estimation and inferences for high‐dimensional linear models. By smoothing over partial regression estimates based on a given variable selection scheme, we reduce the problem to low‐dimensional least squares estimations. The procedure, termed as Selection‐assisted Partial Regression and Smoothing (SPARES), utilizes data splitting along with variable selection and partial regression. We show that the SPARES estimator is asymptotically unbiased and normal, and derive its variance via a nonparametric delta method. The utility of the procedure is evaluated under various simulation scenarios and via comparisons with the de‐biased LASSO estimators, a major competitor. We apply the method to analyze two genomic datasets and obtain biologically meaningful results.