Low-rank modeling for functional linear regression with two-way functional predictor

In this paper, we study a flexible functional linear regression model where the dependency of a scalar response on a functional predictor is function-valued process rather than conventional one-way processes. Additionally, we provide an intuitively appealing estimation approach to estimate the bivariate functional regression coefficient. We first represent the bivariate functional coefficient by using the data-driven bases function to achieve the dimension reduction, and then introduce an iterative least square procedure to estimate the coefficients after dimension reduction in the framework of low-rank structure. Theoretically, we investigate the convergence rate of bivariate functional coefficient estimator under mild conditions. Simulation studies indicate that the proposed methods perform well in finite samples and an empirical example is presented to illustrate its usefulness.