Abstract: We study the simultaneous domain selection problem for varying coefficient models as a functional regression model for longitudinal data with many covariates. The domain selection problem in a functional regression mostly appears within a functional linear regression with a scalar response; however, there is no direct correspondence to functional response models with many covariates. We reformulate the problem as a nonparametric function estimation problem under the notion of functional sparsity. Sparsity encapsulates interpretability in a regression with multiple inputs, and the problem of sparse estimation is well understood in the context of variable selection in a parametric setting. For nonparametric models, interpretability not only concerns the number of covariates involved, but also the zero regions in the functional form. Thus, the sparsity consideration is much more complex. To distinguish the types of sparsity in nonparametric models, we refer to the former as global sparsity and to the latter as local sparsity, both of which constitute functional sparsity. Most existing methods focus on directly extending the framework of parametric sparsity for linear models to nonparametric models to address one type of sparsity, but not both. We develop a penalized estimation procedure that simultaneously addresses both types of sparsity in a unified framework. We establish the asymptotic properties of estimation consistency and sparsistency of the proposed method. Our method is illustrated by means of a simulation study and real-data analysis, and is shown to outperform existing methods in terms of identifying both local and global sparsity.

Key words and phrases: Functional sparsity, group bridge, longitudinal data, model selection, nonparametric regression.