Local Linear Regression for Functional Ergodic Data with Missing at Random Responses

In this article, we develop a novel kernel-based estimation framework for functional regression models in the presence of missing responses, with particular emphasis on the Missing At Random (MAR) mechanism. The analysis is carried out in the setting of stationary and ergodic functional data, where we introduce apparently for the first time a local linear estimator of the regression operator. The principal theoretical contributions of the paper may be summarized as follows. First, we establish almost sure uniform rates of convergence for the proposed estimator, thereby quantifying its asymptotic accuracy in a strong sense. Second, we prove its asymptotic normality, which provides the foundation for distributional approximations and subsequent inference. Third, we derive explicit closed-form expressions for the associated asymptotic variance, yielding a precise characterization of the limiting law. These results are obtained under standard structural assumptions on the relevant functional classes and under mild regularity conditions on the underlying model, ensuring broad applicability of the theory. On the methodological side, the asymptotic analysis is exploited to construct pointwise confidence regions for the regression operator, thereby enabling valid statistical inference. Furthermore, a comprehensive set of simulation experiments is conducted, demonstrating that the proposed estimator exhibits superior finite-sample predictive performance when compared to existing procedures, while simultaneously retaining robustness in the presence of missingness governed by MAR mechanisms.