Nonparametric Functional Times Series Data Analysis by kNN–Local Linear M-Regression

This paper addresses the problem of nonparametric regression for functional time series, a setting complicated by the infinite-dimensional nature of the covariates, temporal dependence, and potential for outliers. We propose a new robust estimator that combines three powerful ideas: (i) k-nearest neighbors (kNN) for adaptive localization in the functional space; (ii) local linear smoothing to reduce bias; and (iii) M-estimation to ensure resilience against atypical observations. The key theoretical contribution establishes the almost-complete convergence of the proposed estimator under mild conditions that account for the functional geometry, weak dependence (via quasi-association), and robustness constraints. The obtained rate of convergence explicitly reveals the interplay between the functional concentration, dependence strength, and local smoothness of the model. A simulation study demonstrates that this method offers superior stability and predictive accuracy compared to classical alternatives, particularly under heavy-tailed errors and data contamination. The practical relevance of the approach is further illustrated through a one-step-ahead prediction application to a real-world environmental dataset of hourly NOx measurements.