<i>k</i>-Nearest Neighbour Estimation of the Conditional Set-Indexed Empirical Process for Functional Data: Asymptotic Properties
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 <i>k</i>-nearest neighbor approach to estimate the regression function, as oppos...
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2025-01-01
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| author | Youssouf Souddi Salim Bouzebda |
| author_facet | Youssouf Souddi Salim Bouzebda |
| author_sort | Youssouf Souddi |
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| description | 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 <i>k</i>-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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">C</mi></semantics></math></inline-formula>, 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 <i>k</i>-nearest neighbor method outperforms the classical approach due to several advantages. |
| format | Article |
| id | doaj-art-681bfdc2321f41c09d25100be8fe8a5e |
| institution | DOAJ |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
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| spelling | doaj-art-681bfdc2321f41c09d25100be8fe8a5e2025-08-20T03:12:12ZengMDPI AGAxioms2075-16802025-01-011427610.3390/axioms14020076<i>k</i>-Nearest Neighbour Estimation of the Conditional Set-Indexed Empirical Process for Functional Data: Asymptotic PropertiesYoussouf Souddi0Salim Bouzebda1Laboratory of Stochastic Models, Statistics and Applications, University of Saida-Dr. Moulay Tahar, P.O. Box 138 EN-NASR, Saïda 20000, AlgeriaLaboratoire de Mathématiques Appliquées de Compiègne (L.M.A.C.), Université de Technologie de Compiègne, 60200 Compiègne, FranceThe 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 <i>k</i>-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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">C</mi></semantics></math></inline-formula>, 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 <i>k</i>-nearest neighbor method outperforms the classical approach due to several advantages.https://www.mdpi.com/2075-1680/14/2/76conditional distributionsmall ball probabilityempirical processmixing functional datasemi-metric spacecovering number |
| spellingShingle | Youssouf Souddi Salim Bouzebda <i>k</i>-Nearest Neighbour Estimation of the Conditional Set-Indexed Empirical Process for Functional Data: Asymptotic Properties Axioms conditional distribution small ball probability empirical process mixing functional data semi-metric space covering number |
| title | <i>k</i>-Nearest Neighbour Estimation of the Conditional Set-Indexed Empirical Process for Functional Data: Asymptotic Properties |
| title_full | <i>k</i>-Nearest Neighbour Estimation of the Conditional Set-Indexed Empirical Process for Functional Data: Asymptotic Properties |
| title_fullStr | <i>k</i>-Nearest Neighbour Estimation of the Conditional Set-Indexed Empirical Process for Functional Data: Asymptotic Properties |
| title_full_unstemmed | <i>k</i>-Nearest Neighbour Estimation of the Conditional Set-Indexed Empirical Process for Functional Data: Asymptotic Properties |
| title_short | <i>k</i>-Nearest Neighbour Estimation of the Conditional Set-Indexed Empirical Process for Functional Data: Asymptotic Properties |
| title_sort | i k i nearest neighbour estimation of the conditional set indexed empirical process for functional data asymptotic properties |
| topic | conditional distribution small ball probability empirical process mixing functional data semi-metric space covering number |
| url | https://www.mdpi.com/2075-1680/14/2/76 |
| work_keys_str_mv | AT youssoufsouddi ikinearestneighbourestimationoftheconditionalsetindexedempiricalprocessforfunctionaldataasymptoticproperties AT salimbouzebda ikinearestneighbourestimationoftheconditionalsetindexedempiricalprocessforfunctionaldataasymptoticproperties |