Asymmetric Estimation for Varying-Coefficient Additive Model with Functional Response in Reproducing Kernel Hilbert Space

Function-on-scalar regression models are extensively utilized in applications involving longitudinal or functional responses. Prior literature has established the minimax optimal bounds for both mean and quantile regression. This paper explores expectile regression as a natural extension to mean regression, particularly for modeling potential heteroscedasticity in data. We propose an expectile function-on-scalar regression model that focuses on asymmetrical regression of functional responses based on scalar predictors. Employing the structure of Reproducing Kernel Hilbert Space (RKHS), we have developed a statistically efficient expectile estimator. This estimator comes with theoretical backing, derived from the minimax rates of convergence in both random and fixed design contexts. Our extensive simulations demonstrate the robust performance of the proposed methods across various settings. Additionally, we present an empirical analysis using quality of life data from a breast cancer clinical trial, showcasing the practical utility of our method.