On Function-on-Scalar Quantile Regression

Abstract

Existing work on functional response regression has focused predominantly on mean regression. However, in many applications, predictors may not strongly influence the conditional mean of functional responses, but other characteristics of their conditional distribution. In this paper, we study function-on-scalar quantile regression, or functional quantile regression (FQR), which can provide a comprehensive understanding of how scalar predictors influence the conditional distribution of functional responses. We introduce a scalable, distributed strategy to perform FQR that can account for intrafunctional dependence structures in the functional responses. This general distributed strategy first performs separate quantile regression to compute $M$-estimators at each sampling location, and then carries out estimation and inference for the entire coefficient functions by properly exploiting the uncertainty quantifications and dependence structures of $M$-estimators. We derive a uniform Bahadur representation and a strong Gaussian approximation result for the $M$-estimators on the discrete sampling grid, which are of independent interest and provide theoretical justification for this distributed strategy. Some large sample properties of the proposed coefficient function estimators are described. Interestingly, our rate calculations show a phase transition phenomenon that has been previously observed in functional mean regression. We conduct simulations to assess the finite sample performance of the proposed methods, and present an application to a mass spectrometry proteomics dataset, in which the use of FQR to delineate the relationship between functional responses and predictors is strongly warranted