Nonparametric Functional Least Absolute Relative Error Regression: Application to Econophysics

Abstract

In this paper, we propose an alternative kernel estimator for the regression operator of scalar response variable S given a functional random variable T that takes values in a semi-metric space. The new estimator is constructed through the minimization of the least absolute relative error (LARE). The latter is characterized by its ability to provide a more balanced and scale-invariant measure of prediction accuracy compared to traditional standard absolute or squared error criterion. The LARE is an appropriate tool for reducing the influence of extremely large or small response values, enhancing robustness against heteroscedasticity or/and outliers. This feature makes the LARE suitable for functional or high-dimensional data, where variations in scale response are common. The high feasibility and strong performance of the proposed estimator are supported theoretically by establishing its stochastic consistency. The latter is derived with precision of the convergence rate under mild regularity conditions. The ease implementation and the stability of the estimator are justified by simulation studies and an empirical application to near-infrared (NIR) spectrometry data. Of course, to explore the functional architecture of this data, we employ random matrix theory (RMT), which is a principal analytical tool of econophysics.