Asymptotic normality of the k-NN single index regression estimator for functional weak dependence data*

Mohammedi, Mustapha; Bouzebda, Salim; Laksaci, Ali; Bouanani, Oussama

doi:10.1080/03610926.2022.2150823

Cited by 6 publications

(1 citation statement)

References 42 publications

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“…In [25], the authors examine the application of the kernel nearest neighbors (k-nn) technique in a regression model with a single index. They specifically focus on cases where the explanatory variable is measured in functional space, in the context of the association dependency condition.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Results of Some Conditional Nonparametric Functional Parameters in High-Dimensional Associated Data

Daoudi,

Elmezouar,

Alshahrani

2023

Mathematics

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In this paper, we propose to study the asymptotic properties of some conditional functional parameters, such as the distribution function, the density, and the hazard function, for an explanatory variable with values in a Hilbert space (infinite dimension) and a response variable real in a quasi-associated dependency framework. We consider the non parametric estimation of the conditional distribution function by the kernel method in the presence of the quasi-associated dependence, and we establish under general hypotheses the almost complete convergence with speed of the estimator built in the associated case. The estimation of the conditional hazard function will be conducted by utilizing the two outcomes of the conditional distribution function and the conditional density. We establish the asymptotic normality of the kernel estimator as the conditional risk function of a properly normalized functional. We explicitly give the asymptotic variance. Simulation studies were conducted to investigate the behavior of the asymptotic property in the context of finite sample data. All the statistical analyses were performed using R software.

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Section: Introductionmentioning

confidence: 99%

Asymptotic Results of Some Conditional Nonparametric Functional Parameters in High-Dimensional Associated Data

Daoudi,

Elmezouar,

Alshahrani

2023

Mathematics

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Weak convergence of the conditional U-statistics for locally stationary functional time series

Soukarieh,

Bouzebda

2023

Stat Inference Stoch Process

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Uniform in Number of Neighbor Consistency and Weak Convergence of k-Nearest Neighbor Single Index Conditional Processes and k-Nearest Neighbor Single Index Conditional U-Processes Involving Functional Mixing Data

Bouzebda

2024

Symmetry

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U-statistics are fundamental in modeling statistical measures that involve responses from multiple subjects. They generalize the concept of the empirical mean of a random variable X to include summations over each m-tuple of distinct observations of X. W. Stute introduced conditional U-statistics, extending the Nadaraya–Watson estimates for regression functions. Stute demonstrated their strong pointwise consistency with the conditional expectation r(m)(φ,t), defined as E[φ(Y1,…,Ym)|(X1,…,Xm)=t] for t∈Xm. This paper focuses on estimating functional single index (FSI) conditional U-processes for regular time series data. We propose a novel, automatic, and location-adaptive procedure for estimating these processes based on k-Nearest Neighbor (kNN) principles. Our asymptotic analysis includes data-driven neighbor selection, making the method highly practical. The local nature of the kNN approach improves predictive power compared to traditional kernel estimates. Additionally, we establish new uniform results in bandwidth selection for kernel estimates in FSI conditional U-processes, including almost complete convergence rates and weak convergence under general conditions. These results apply to both bounded and unbounded function classes, satisfying certain moment conditions, and are proven under standard Vapnik–Chervonenkis structural conditions and mild model assumptions. Furthermore, we demonstrate uniform consistency for the nonparametric inverse probability of censoring weighted (I.P.C.W.) estimators of the regression function under random censorship. This result is independently valuable and has potential applications in areas such as set-indexed conditional U-statistics, the Kendall rank correlation coefficient, and discrimination problems.

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Asymptotic normality of the k-NN single index regression estimator for functional weak dependence data*

Cited by 6 publications

References 42 publications

Asymptotic Results of Some Conditional Nonparametric Functional Parameters in High-Dimensional Associated Data

Asymptotic Results of Some Conditional Nonparametric Functional Parameters in High-Dimensional Associated Data

Weak convergence of the conditional U-statistics for locally stationary functional time series

Uniform in Number of Neighbor Consistency and Weak Convergence of k-Nearest Neighbor Single Index Conditional Processes and k-Nearest Neighbor Single Index Conditional U-Processes Involving Functional Mixing Data

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