Additive partially linear models for ultra‐high‐dimensional regression

Abstract

We consider a semiparametric additive partially linear regression model (APLM) for analysing ultra‐high‐dimensional data where both the number of linear components and the number of non‐linear components can be much larger than the sample size. We propose a two‐step approach for estimation, selection, and simultaneous inference of the components in the APLM. In the first step, the non‐linear additive components are approximated using polynomial spline basis functions, and a doubly penalized procedure is proposed to select nonzero linear and non‐linear components based on adaptive lasso. In the second step, local linear smoothing is then applied to the data with the selected variables to obtain the asymptotic distribution of the estimators of the nonparametric functions of interest. The proposed method selects the correct model with probability approaching one under regularity conditions. The estimators of both the linear part and the non‐linear part are consistent and asymptotically normal, which enables us to construct confidence intervals and make inferences about the regression coefficients and the component functions. The performance of the method is evaluated by simulation studies. The proposed method is also applied to a dataset on the shoot apical meristem of maize genotypes.