Robust Function-on-Function Regression: A Penalized Tau-based Estimation Approach

This study introduces a novel penalized estimation method tailored for function-on-function regression models, combining the robustness of the Tau estimator with penalization techniques to enhance resistance to outliers. Function-on-function regression is essential for modeling intricate relationships between functional predictors and response variables across diverse fields. However, traditional methods often struggle with outliers, leading to biased estimates and diminished predictive performance. Our proposed approach addresses this challenge by integrating robust Tau estimation with penalization, promoting both robustness and parsimony in parameter estimation. Theoretical foundations of the penalized Tau estimator within function-on-function regression are discussed, along with empirical validations through simulation studies and an empirical data analysis. By incorporating penalization, our method not only ensures robust estimation of regression parameters but also promotes model simplicity, offering enhanced interpretability and generalization capabilities in functional data analysis.