The functional linear model is an important extension of the classical regression model allowing for scalar responses to be modeled as functions of stochastic processes. Yet, despite the usefulness and popularity of the functional linear model in recent years, most treatments, theoretical and practical alike, suffer either from (i) lack of resistance towards the many types of anomalies one may encounter with functional data or (ii) biases resulting from the use of discretely sampled functional data instead of completely observed data. To address these deficiencies, this paper introduces and studies the first class of robust functional regression estimators for partially observed functional data. The proposed broad class of estimators is based on thin-plate splines with a novel computationally efficient quadratic penalty, is easily implementable and enjoys good theoretical properties under weak assumptions. We show that, in the incomplete data setting, both the sample size and discretization error of the processes determine the asymptotic rate of convergence of functional regression estimators and the latter cannot be ignored. These theoretical properties remain valid even with multi-dimensional random fields acting as predictors and random smoothing parameters. The effectiveness of the proposed class of estimators in practice is demonstrated by means of a simulation study and a real-data example.

翻译：函数型线性模型是经典回归模型的重要扩展，允许将标量响应建模为随机过程的函数。尽管函数型线性模型近年来具有实用性和普及性，但大多数理论和实践研究要么（i）缺乏对函数型数据中可能出现的多种异常值的抵抗能力，要么（ii）因使用离散采样函数型数据而非完全观测数据而产生偏差。为解决这些缺陷，本文首次提出并研究了一类针对部分观测函数型数据的稳健函数型回归估计量。该宽泛估计量类基于薄板样条函数，并结合了一种新颖的计算高效的二次惩罚项，易于实现且在弱假设下具有良好的理论性质。我们证明，在不完全数据设定中，样本量与过程的离散化误差共同决定了函数型回归估计量的渐近收敛速率，且后者不可忽略。即使预测变量为多维随机场且平滑参数为随机参数，这些理论性质依然成立。通过模拟研究与实际数据案例，验证了所提估计量类在实践中的有效性。