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

Abstract

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.functional data, penalized regression, robust procedures | INTRODUCTIONAdvancements in data collection and storage technology have enabled experimenters to collect complex and high-dimensional datasets, such as curves, images or other objects over multiple time or spatial points. This type of data is called functional data and has been encountered frequently in many scientific fields such as ecology, geophysics and medical sciences; see Ramsay and Silverman (2005), Ferraty and Vieu (2006), Horváth and Kokoszka (2012) and Zhu et al. (2012 for a general overview of functional data analysis (FDA).

Sparse estimation of historical functional linear models with a nested group bridge approach

“…Cai, Xue & Cao (2020) proposed a variable selection procedure for the function‐on‐function regression model using the group smoothly clipped absolute deviation regulation method. Cai, Xue & Cao (2021) presented a robust method for the function‐on‐function regression model using M‐estimation and penalized spline regression.…”