Shrinkage estimation and selection for multiple functional regression

Functional linear regression is a useful extension of simple linear regression and has been investigated by many researchers. However, the functional variable selection problem when multiple functional observations exist, which is the counterpart in the functional context of multiple linear regression, is seldom studied. Here we propose a method using a group smoothly clipped absolute deviation penalty (gSCAD) which can perform regression estimation and variable selection simultaneously. We show the method can identify the true model consistently, and discuss construction of pointwise confidence intervals for the estimated functional coefficients. Our methodology and theory is verified by simulation studies as well as some applications to data.