Empirical likelihood M-estimation for the varying-coefficient model with functional response

This work is motivated by a gap in the functional data analysis literature, particularly in the context of neuroimaging, regarding the ability of functional models to robustly accommodate intra-observation dependence. In response, we propose an M-estimator based on generalized empirical likelihood for the varying-coefficient model with a functional response. We develop statistical inference procedures, simultaneous confidence regions, and a global general linear hypothesis test for the model's functional coefficient. Our theoretical results establish the weak convergence of the log-likelihood ratio process, a nonparametric version of Wilks' theorem for the log-likelihood ratio, and asymptotic properties of the proposed estimator. Through a simulation study, we show that the proposed confidence sets have close-to-nominal coverage probabilities. In a real-world application to a neuroimaging dataset, we show that mini-mental state examination score and apolipoprotein E genotype have significant associations with fractional anisotropy, while associations with gender and age are only present at high quantile levels.