Abstract: We propose a new functional response quantile regression model, and develop a data-driven estimation procedure to estimate the quantile regression processes based on a local linear approximation. Theoretically, we obtain the global uniform Bahadur representation of the estimator with respect to the time/location and the quantile level, and show that the estimator converges weakly to a two-parameter continuous Gaussian process. We then derive the asymptotic bias and mean integrated squared error of the smoothed individual functions and their uniform convergence rates under given quantile levels. Based on the theoretical results, we introduce a global test for the coefficient functions and discuss how to construct simultaneous confidence bands. We evaluate our method using simulations and by applying it to diffusion tensor imaging data and ADHD-200 functional magnetic resonance imaging data.

Key words and phrases: Functional data, global test statistic, simultaneous confidence band, weak convergence.