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 or independent assumptions on the residual process, with the focus on statistical inference and 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, 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.

翻译：函数分位数回归（FQR）是函数型数据均值回归的一种有效替代方法，因为它能全面理解标量预测变量如何影响函数型响应的条件分布。本文研究了密集采样、高维函数型数据的FQR模型，不依赖残差过程的参数假设或独立性假设，重点关注统计推断和可扩展实现。这是通过一种简单而强大的分布式策略实现的：首先在每个采样位置独立进行分位数回归，计算$M$估计量；然后通过合理利用$M$估计量的不确定性量化和依赖结构，对整个系数函数进行估计和推断。我们推导了离散采样网格上$M$估计量的一致Bahadur表示和强高斯近似结果，作为推断的基础。提出了具有极小极大最优性的插值估计量，并建立了点和同时区间估计的大样本性质。在FQR模型下获得的极小极大最优率揭示了一个有趣的相变现象，该现象此前已在函数型均值回归中观察到。通过模拟实验和一个质谱蛋白质组学数据集的实例应用对所提方法进行了验证。