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.

翻译：函数分位数回归（FQR）是函数数据均值回归的一种有效替代方法，因为它能够全面理解标量预测变量如何影响函数响应的条件分布。本文研究了针对稠密采样、高维函数数据的FQR模型，既不依赖参数误差假设，也不依赖独立随机过程假设，重点关注这一具有挑战性场景下的统计推断及其可扩展实现。这通过一种简单而强大的分布式策略实现：我们首先在每个采样位置分别进行分位数回归以计算M估计量，然后通过恰当利用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估计量的统一Bahadur表示和强高斯逼近结果，从而实现了降维并作为推断的基础。提出了一种具有极小极大最优性的基于插值的估计量，并建立了点估计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速率展现了一个有趣的相变现象，该现象先前已在函数均值回归中观察到。通过模拟实验和质谱蛋白质组学数据集的应用对所提方法进行了验证。