Estimation and inference in generalized additive coefficient models for nonlinear interactions with high-dimensional covariates

TL;DR

This paper develops estimation and inference methods for high-dimensional generalized additive coefficient models, enabling analysis of nonlinear interactions with many covariates, validated through simulations and real data application.

Contribution

It introduces a groupwise penalization approach for model selection and confidence band construction in high-dimensional GACMs, extending existing low-dimensional methods.

Findings

Consistent model structure identification in high dimensions.

Effective construction of simultaneous confidence bands.

Good numerical performance demonstrated in simulations and real data.

Abstract

In the low-dimensional case, the generalized additive coefficient model (GACM) proposed by Xue and Yang [Statist. Sinica 16 (2006) 1423-1446] has been demonstrated to be a powerful tool for studying nonlinear interaction effects of variables. In this paper, we propose estimation and inference procedures for the GACM when the dimension of the variables is high. Specifically, we propose a groupwise penalization based procedure to distinguish significant covariates for the "large $p$ small $n$" setting. The procedure is shown to be consistent for model structure identification. Further, we construct simultaneous confidence bands for the coefficient functions in the selected model based on a refined two-step spline estimator. We also discuss how to choose the tuning parameters. To estimate the standard deviation of the functional estimator, we adopt the smoothed bootstrap method. We conduct simulation experiments to evaluate the numerical performance of the proposed methods and analyze an obesity data set from a genome-wide association study as an illustration.