Uniform in Bandwidth Consistency of the L¹-Modal Regression Estimator for High-Dimensional Data

We propose a new nonparametric estimator of the conditional mode in a regression framework where the covariates are functional in nature. The estimator is constructed through a quantile regression approach, which provides a robust alternative to classical density-based procedures. It is well documented that employing the L1-structure in quantile regression, the estimation procedure improves robustness properties, particularly resistance to outliers and heavy-tailed error distributions. This feature makes the L1 estimation of the conditional mode more stable and reliable in complex and high-variability functional data settings. The main objective of this paper is to establish strong consistency, with explicit convergence rates, for the associated kernel estimators, uniformly over a range of bandwidth parameters. The latter is developed under general regularity conditions involving the concentration distribution of the functional regressor, smoothness assumptions on the structural components of the model, and entropy conditions ensuring adequate control of the functional class complexity. Uniformity in bandwidth is essential both from a theoretical and practical issues, as it guarantees stability of the estimator under data-driven smoothing parameter selection. Beyond its theoretical contribution, this paper has direct implications for applied statistics. Specifically, it provides mathematical support for the automatic bandwidth selection procedures in the high-dimensional data context. Furthermore, the main theoretical novelty is highlighted through simulation experiments and applications to real data.