M-estimation for varying coefficient models with a functional response in a reproducing kernel Hilbert space

Modern neuroimaging research calls for statistical methods that can model dynamic relationships between a functional response and a set of covariates. Current methods, however, remain disparate and limited in their ability to robustly accommodate real-world data and integrate smoothness penalties. In this work, we propose an M-estimation framework for the varying-coefficient model with a functional response that encompasses both mean and quantile regression. To accommodate smoothness regularization and circumvent the stringent conditions on Fourier coefficients or the covariance operator's eigenvalues imposed by traditional fixed-basis representations, we assume that the functional coefficient resides in a reproducing kernel Hilbert space. We show that our proposed estimator is minimax rate optimal and establish convergence properties of our modified alternating direction method of multipliers algorithm. We further propose combining a weighted M-estimator and a copula model to quantify within-subject spatial dependence to improve estimation accuracy. Simulation studies and a real-world analysis demonstrate the robustness of our proposed methods to outliers.