Asymptotic Distribution of the Functional Modal Regression Estimator

We propose a novel predictor for functional time series (FTS) based on the robust estimation of the modal regression within a functional statistics framework. The robustness of the estimator is incorporated through the L1-estimation of the quantile density. Such consideration improves the precision of conditional mode estimation. A principal theoretical contribution of this work is the establishment of the asymptotic normality of the proposed estimator. This result is of considerable importance, as it provides the foundation for statistical inference, including hypothesis testing and the construction of confidence intervals. Therefore, the obtained asymptotic result enhances the practical usability of the modal regression prediction. On the empirical side, we evaluate the performance of the estimator under various smoothing structures using both simulated and real data. The real data application highlights the ability of the L1-conditional mode predictor to perform robust and reliable short-term forecasts, with very high effectiveness in the analysis of economic data.