{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T00:37:50Z","timestamp":1759970270564,"version":"build-2065373602"},"reference-count":88,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,1,21]],"date-time":"2025-01-21T00:00:00Z","timestamp":1737417600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>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 C, 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.<\/jats:p>","DOI":"10.3390\/axioms14020076","type":"journal-article","created":{"date-parts":[[2025,1,21]],"date-time":"2025-01-21T05:47:42Z","timestamp":1737438462000},"page":"76","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["k-Nearest Neighbour Estimation of the Conditional Set-Indexed Empirical Process for Functional Data: Asymptotic Properties"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0009-0003-9666-6076","authenticated-orcid":false,"given":"Youssouf","family":"Souddi","sequence":"first","affiliation":[{"name":"Laboratory of Stochastic Models, Statistics and Applications, University of Saida-Dr. Moulay Tahar, P.O. Box 138 EN-NASR, Sa\u00efda 20000, Algeria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7801-4945","authenticated-orcid":false,"given":"Salim","family":"Bouzebda","sequence":"additional","affiliation":[{"name":"Laboratoire de Math\u00e9matiques Appliqu\u00e9es de Compi\u00e8gne (L.M.A.C.), Universit\u00e9 de Technologie de Compi\u00e8gne, 60200 Compi\u00e8gne, France"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Pollard, D. (1990). Empirical Processes: Theory and Applications, American Statistical Association. NSF-CBMS Regional Conference Series in Probability and Statistics.","DOI":"10.1214\/cbms\/1462061091"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Shorack, G.R., and Wellner, J.A. (2009). Empirical Processes with Applications to Statistics, Society for Industrial and Applied Mathematics (SIAM). 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