Estimation and inference in generalized additive coefficient models for nonlinear interactions with high-dimensional covariates
Shujie Ma, Raymond J. Carroll, Hua Liang, Shizhong Xu
The Annals of Statistics · 2015 · Volume 43 · Issue 5
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.
Keywords
62F12, 62G08, 62G10, 62G20, 62J02, adaptive group LASSO, bootstrap smoothing, curse of dimensionality, gene-environment interaction, generalized additive partially linear models, inference for high-dimensional data, oracle property, penalized likelihood, polynomial splines, two-step estimation, undersmoothing
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