Abstract

In this paper, we propose a new nonparametric method for estimating the regression operator of a scalar response given a functional covariate taking values in a semi-metric space. The estimator is obtained by minimizing the Least Absolute Relative Error (LARE) criterion, which provides a scale-invariant and equilibrated measure of prediction accuracy compared with classical regression loss functions. The antisymmetry property of the LARE rule ensures that overestimation and underestimation are penalized in a symmetric relative manner, which improves the robustness when the response variable varies in different scales. Next, the estimator is constructed using k-nearest neighbors (kNN). The combination of the two algorithms allows the procedure to benefit from both the robustness and scale-invariant nature of the LARE criterion and the flexibility and local adaptivity of the kNN smoothing approach, which is particularly suitable for functional or high-dimensional data. As an asymptotic result, we establish the uniform convergence with respect to the number of neighbors (UNN) of the proposed estimator under mild regularity conditions and derive its rate of convergence. We also discuss the selection of the optimal number of neighbors and their impact on performance. The practical effectiveness of the proposed kNN–FLARE regression estimator is illustrated through simulation experiments and an application to near-infrared (NIR) spectrometry data.

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