Scalar on Function Absolute Relative Error Regression: Functional Time Series Case

This paper introduces a novel estimator for the functional regression model with a scalar response variable D and a functional predictor $\mathcal{C}$ taking values in a semi-metric space. The proposed estimator is obtained by minimizing the Least Absolute Relative Error (LARE) loss, an asymmetric criterion that measures prediction errors relative to the magnitude of the response variable. The theoretical properties of the estimator are established in the functional time series case by deriving its asymptotic distribution. This result provides a fundamental basis for some statistical inferences, including the construction of confidence intervals and the development of hypothesis tests for the regression operator. Unlike the conventional least absolute deviation or least squares criteria, the absolute relative error criterion offers a scale-invariant measure of prediction accuracy, making it particularly suitable when the response variable exhibits substantial variability. By the relative absolute deviations, the proposed approach reduces the impact of extreme observations, improves robustness to heteroscedasticity and outliers. Thus, approach enhances prediction performance in functional time series. The practical relevance of the proposed methodology is highlighted through extensive simulation experiments and an application to a real-world dataset, proving its computational simplicity, stability and accuracy.