Abstract

Estimation and prediction in functional linear models with scalar response and fully or densely observed functional covariates have been widely studied in the literature. However, the extension to sparsely observed functional covariates remains an open problem. The key obstacle in prediction is the approximation of the index in the functional linear model, which involves the inner product of the functional covariates with the corresponding coefficient function. However, such an approximation cannot be consistent if the functional covariates are observed sparsely at only a few time points. We provide a solution by imputing the sample functional path and substituting the imputed path into the original functional linear model. We show why such a substitution method works and propose two estimation approaches: one through the normal equation, which targets the population quantities, and the other through estimation of the slope function using a reproducing kernel Hilbert space approach. Asymptotic properties of these estimators and their corresponding prediction methods are established. Numerical performance of the proposed methods is evaluated through simulations and real data from the Framingham Heart Study.