  Functional quantile regression (FQR) is a useful alternative to mean
regression for functional data as it provides a comprehensive understanding of
how scalar predictors influence the conditional distribution of functional
responses. In this article, we study the FQR model for densely sampled,
high-dimensional functional data without relying on parametric error or
independent stochastic process assumptions, with the focus on statistical
inference under this challenging regime along with scalable implementation.
This is achieved by a simple but powerful distributed strategy, in which we
first perform separate quantile regression to compute $M$-estimators at each
sampling location, and then carry out estimation and inference for the entire
coefficient functions by properly exploiting the uncertainty quantification and
dependence structure of $M$-estimators. We derive a uniform Bahadur
representation and a strong Gaussian approximation result for the
$M$-estimators on the discrete sampling grid, leading to dimension reduction
and serving as the basis for inference. An interpolation-based estimator with
minimax optimality is proposed, and large sample properties for point and
simultaneous interval estimators are established. The obtained minimax optimal
rate under the FQR model shows an interesting phase transition phenomenon that
has been previously observed in functional mean regression. The proposed
methods are illustrated via simulations and an application to a mass
spectrometry proteomics dataset.
