Estimation of Functional Sparsity in Nonparametric Varying Coefficient Models for Longitudinal Data Analysis

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 functional regression mostly appears under the functional linear regression with scalar response, but there is no direct correspondence to functional response models with many covariates. We reformulate the problem as nonparametric function estimation under the notion of functional sparsity. Sparsity is the recurrent theme that encapsulates interpretability in the face of regression with multiple inputs, and the problem of sparse estimation is well understood in the parametric setting as variable selection. For nonparametric models, interpretability not only concerns the number of covariates involved but also the functional form of the estimates, and so the sparsity consideration is much more complex. To distinguish the types of sparsity in nonparametric models, we call the former global sparsity and the latter local sparsity, which constitute functional sparsity. Most existing methods focus on directly extending the framework of parametric sparsity for linear models to nonparametric function estimation to address one or the other, but not both. We develop a penalized estimation procedure that simultaneously addresses both types of sparsity in a unified framework. We establish asymptotic properties of estimation consistency and sparsistency of the proposed method. Our method is illustrated in simulation study and real data analysis, and is shown to outperform the existing methods in identifying both local sparsity and global sparsity.