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

• Activity Number: 27
• Type: Contributed
• Date/Time: Sunday, August 3, 2014 : 2:00 PM to 3:50 PM
• Sponsor: SSC
• Author(s): Zhenhua Lin*+ and Jiguo Cao and Liangliang Wang and Haonan Wang
• Companies: University of Toronto and Simon Fraser University and Simon Fraser University and Colorado State University
• Keywords: regularization ; null region ; oracle property ; smoothly clipped absolute deviation

Abstract

In this paper, we first propose a new regularization technique called "functional SCAD", and then combine it 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 sub-regions 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 constraining the number of knots. Compared to other available methods, our method is theoretically sounder and computationally simpler. An asymptotic analysis shows that our estimator enjoys the oracle property, which means asymptotically it performs as well as if we knew the true null sub-regions of the coefficient function. Furthermore, simulation studies show that our estimator has superior numerical performance. Finally, the practical merit of our method is demonstrated on the classic Canadian weather data.