The Annals of Statistics
- Ann. Statist.
- Volume 43, Number 5 (2015), 2102-2131.
Estimation and inference in generalized additive coefficient models for nonlinear interactions with high-dimensional covariates
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.
Ann. Statist., Volume 43, Number 5 (2015), 2102-2131.
Received: September 2014
Revised: May 2015
First available in Project Euclid: 3 August 2015
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Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
Ma, Shujie; Carroll, Raymond J.; Liang, Hua; Xu, Shizhong. Estimation and inference in generalized additive coefficient models for nonlinear interactions with high-dimensional covariates. Ann. Statist. 43 (2015), no. 5, 2102--2131. doi:10.1214/15-AOS1344. https://projecteuclid.org/euclid.aos/1438606855
- Supplemental materials for “Estimation and inference in generalized additive coefficient models for nonlinear interactions with high-dimensional covariates”. The supplementary material presents additional numerical results and the proofs of Lemmas A.1 and A.2.