Asymptotic distribution of functional least absolute relative error regression

Abstract This paper introduces an alternative estimator for the functional regression with a scalar response variable S given a functional random variable T $\mathcal{T}$ taking values in a semi-metric space. The estimator is obtained by minimizing the least absolute relative error (LARE). Unlike conventional absolute or squared error criteria, the LARE provides an equilibrium scale-invariant measure of prediction accuracy. It permits to increase the efficiency of the predictor by reducing the influence of extreme values and enhancing robustness against heteroscedasticity and outliers. These characteristics render the LARE more adequate for functional and high-dimensional data, where scale variation is often observed. The theoretical validity of the proposed estimator is stated through the establishment of the asymptotic normality of the constructed estimator. The latter provides the necessary foundation for constructing confidence intervals and conducting hypothesis testing on the regression operator. The practical utility, implementation ease, and stability of the method are proved through simulation studies and real data application.