Abstract
Scalar-on-function regression allows for a scalar response to be dependent on functional predictors; however, not much work has been done when interaction effects between the functional predictors are included. In this paper, we introduce a multiple functional linear regression model with interaction terms. Meanwhile, we enforce the hierarchical structure constraint on the model, that is, interaction terms can be selected into the model only if the associated main effects are in the model. Based on the functional principal component analysis and group smoothly clipped absolute deviation (SCAD) penalty, we propose a new penalized estimation procedure to select the important functional predictors and interactions while automatically obeying the hierarchical structure. With appropriate selection of the tuning parameters, the rates of convergence of the proposed estimators and the consistency of the model selection procedure are established under some regularity conditions. At last, we illustrate the finite sample performance of our proposed methods with some simulation studies and a real data application.
Funding Statement
Sanying Feng’s research was supported by the Humanities and Social Science Project of Ministry of Education of China (21YJC910003), the Foundation of Henan Educational Committee (No.21A910004), the Natural Science Foundation of Henan Province (No.212300410412).
Lifang Pei’s research was supported by the National Natural Science Foundation of China (No. 11701523).
Citation
Sanying Feng. Xinyu Zhang. Hui Liang. Lifang Pei. "Model selection for functional linear regression with hierarchical structure." Braz. J. Probab. Stat. 36 (2) 243 - 262, June 2022. https://doi.org/10.1214/21-BJPS525
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