Nonparametric Analysis of Functional Time Series Data Using Least Absolute Relative Error Regression

In this paper, we introduce a novel kernel-based estimator for the regression operator of a scalar response variable R given a functional covariate F taking values in a semi-metric space. The estimator is constructed through the minimization of the least absolute relative error (LARE) criterion, which provides an invariant scale and more balanced measure of predictive performance than conventional squared error methods. By focusing on relative deviations, the LARE approach effectively reduces the influence of extreme response values and enhances robustness in the presence of heteroscedasticity. From a theoretical point of view, we investigate the asymptotic behavior of the proposed estimator under strong mixing conditions for functional time series data. We show that, despite the temporal dependence structure, the estimator remains consistent and achieves convergence rates comparable to those obtained under independence. In the computational part, we show that the proposed method is computationally efficient and straightforward to implement. Its empirical performance is evaluated through simulation studies conducted under different dependence scenarios. In addition, the applicability of the method is illustrated through the analysis of a real data set.