Robust penalized M‐estimation for function‐on‐function linear regression

Function‐on‐function linear regression is an essential tool in characterizing the linear relationship between a functional response and a functional predictor. However, most of the estimation methods for this model are based on the least‐squares procedure, which is sensitive to atypical observations. In this paper, we present a robust method for the function‐on‐function linear model using M‐estimation and penalized spline regression. A fast iterative algorithm is provided to compute the estimates. The efficiency of the proposed robust penalized M‐estimator is investigated with several simulation studies in comparison with the conventional method. We demonstrate the performance of the proposed robust method with two real data examples in a capital bike‐sharing study and a Hawaii ocean time‐series program.