面向密集采样函数型数据的可扩展函数对标量分位数回归
Scalable Function-on-Scalar Quantile Regression for Densely Sampled Functional Data
AI总结：
针对密集采样高维函数型数据，提出基于分布式策略的可扩展函数型分位数回归方法，推导相关理论性质并验证其有效性。
摘要：
函数型分位数回归（FQR）是函数型数据均值回归的一种有效替代方法，它能全面揭示标量预测变量如何影响函数型响应的条件分布。本文研究适用于密集采样高维函数型数据的FQR模型，不依赖参数误差或独立随机过程假设，重点关注这一高难度场景下的统计推断及可扩展实现方案。\n我们通过一种简单却高效的分布式策略实现上述目标：首先在每个采样位置分别执行分位数回归以计算$M$-估计量，随后通过合理利用$M$-估计量的不确定性量化结果与依赖结构，对整个系数函数进行估计与推断。我们推导了离散采样网格上$M$-估计量的一致Bahadur表示与强高斯逼近结果，该结果可实现降维，同时是统计推断的基础。本文提出了具有极小极大最优性的基于插值的估计量，并建立了点估计与同时区间估计的大样本性质。FQR模型下得到的极小极大最优速率呈现出一种有趣的相变现象，该现象此前已在函数型均值回归中被观测到。我们通过模拟实验以及在质谱蛋白质组学数据集上的应用，验证了所提方法的有效性。
英文摘要：
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.
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