Time-varying Additive Models for Longitudinal Data

Additive model is an effective dimension reduction model that provides flexibility to model the relation between a response variable and key covariates. The literature is largely developed to scalar response and vector covariates. In this paper, more complex data is of interest, where both the response and covariates may be functions. A functional additive model is proposed together with a new smooth backfitting algorithm to estimate the unknown regression functions, whose components are time-dependent additive functions of the covariates. Due to the sampling plan, such functional data may not be completely observed as measurements may only be collected intermittently at discrete time points. We develop a uniform platform and efficient approach that can cover both dense and sparse functional data and the needed theory for statistical inference. The oracle properties of the component functions are also established.