Scalable Function-on-Scalar Quantile Regression for Densely Sampled Functional Data

TL;DR

This paper develops a scalable, nonparametric functional quantile regression method for densely sampled high-dimensional data, enabling inference without relying on traditional assumptions and demonstrating optimal convergence rates.

Contribution

It introduces a distributed estimation strategy for functional quantile regression that achieves minimax optimality and handles complex dependence structures in high-dimensional functional data.

Findings

Achieves uniform Bahadur representation and Gaussian approximation for estimators.

Proposes an interpolation-based estimator with minimax optimality.

Demonstrates phase transition in convergence rates similar to functional mean regression.

Abstract

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.