A Smooth and Locally Sparse Estimator for Functional Linear Regression via Functional SCAD Penalty

Abstract

In this paper, we propose a new regularization technique called "functional SCAD". We then combine this technique with the smoothing spline method to develop a smooth and locally sparse (i.e., zero on some sub-regions) estimator for the coefficient function in functional linear regression. The functional SCAD has a nice shrinkage property that enables our estimating procedure to identify the null subregions of the coefficient function without over shrinking the non-zero values of the coefficient function. Additionally, the smoothness of our estimated coefficient function is regularized by a roughness penalty rather than by controlling the number of knots. Our method is more theoretically sound and is computationally simpler than the other available methods. An asymptotic analysis shows that our estimator is consistent and can identify the null region with the probability tending to one. Furthermore, simulation studies show that our estimator has superior numerical performance. Finally, the practical merit of our method is demonstrated on two real applications.