k-Nearest Neighbour Estimation of the Conditional Set-Indexed Empirical Process for Functional Data: Asymptotic Properties - Youssouf Souddi; Salim Bouzebda (Axioms)
The main aim of this paper is to improve the existing limit theorems for set-indexed conditional empirical processes involving functional strong mixing random variables. To achieve this, we propose using the k-nearest neighbor approach to estimate the regression function, as opposed to the traditional kernel method. For the first time, we establish the weak consistency, asymptotic normality, and density of the proposed estimator. Our results are derived under certain assumptions about the richness of the index class <semantics>C</semantics>, specifically in terms of metric entropy with bracketing. This work builds upon our previous papers, which focused on the technical performance of empirical process methodologies, and further refines the prior estimator. We highlight that the k-nearest neighbor method outperforms the classical approach due to several advantages.