Functional Linear Regression: Dependence and Error Contamination

Functional linear regression is an important topic in functional data analysis. It is commonly assumed that samples of the functional predictor are independent realizations of an underlying stochastic process, and are observed over a grid of points contaminated by iid measurement errors. In practice, however, the dynamical dependence across different curves may exist and the parametric assumption on the error covariance structure could be unrealistic. In this article, we consider functional linear regression with serially dependent observations of the functional predictor, when the contamination of the predictor by the white noise is genuinely functional with fully nonparametric covariance structure. Inspired by the fact that the autocovariance function of observed functional predictors automatically filters out the impact from the unobservable noise term, we propose a novel autocovariance-based generalized method-of-moments estimate of the slope function. We also develop a nonparametric smoothing approach to handle the scenario of partially observed functional predictors. The asymptotic properties of the resulting estimators under different scenarios are established. Finally, we demonstrate that our proposed method significantly outperforms possible competing methods through an extensive set of simulations and an analysis of a public financial dataset.

函数型线性回归（Functional linear regression）是函数型数据分析（Functional data analysis）中的重要研究方向。通常假设函数型预测变量的样本为某一潜在随机过程的独立实现，并在被独立同分布（iid）测量误差污染的网格点上完成观测。然而在实际应用中，不同函数曲线间可能存在动态相依性，且针对误差协方差结构的参数化假设往往并不切合实际。本文聚焦于函数型预测变量存在序列相依的函数型线性回归问题，此时白噪声对预测变量的污染本身具有函数型形式，且协方差结构为完全非参数形式。受观测函数型预测变量的自协方差函数可自动滤除不可观测噪声项影响这一思路启发，我们提出了一种新颖的基于自协方差的广义矩估计量以估计斜率函数。同时，我们开发了一种非参数平滑方法，用于处理部分观测的函数型预测变量场景。本文建立了不同场景下所得估计量的渐近性质。最后，通过大量模拟实验与公开金融数据集的实证分析，我们证明所提方法显著优于各类现有竞争方法。