Varying-Coefficient Functional Linear Regression

Abstract: Functional linear regression analysis aims to model regression relations which include a functional predictor. The analogue to the regression parameter vector or matrix in conventional multivariate or multiple-response linear regression models is a regression parameter function in one or two arguments. If in addition one has scalar predictors, as is often the case in applications to longitudinal studies, the question arises how to incorporate these into a functional regression model. We study a varying-coefficient approach where the scalar covariates are modeled as additional arguments of the regression parameter function. This extension of the functional linear regression model is analogous to the extension of conventional linear regression models to varying-coefficient models, and shares the advantages such as increased flexibility, however the details of this extension are more challenging in the functional case. Our methodology combines smoothing methods with regularization by truncation at a finite number of functional principal components. A practical version is developed and demonstrated to perform better than functional linear regression for longitudinal data. We investigate the asymptotic properties of varyingcoefficient functional linear regression and establish consistency properties.