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 4 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 neighbors consistency and weak convergence of $ k $NN empirical conditional processes and $ k $NN conditional $ U $-processes involving functional mixing data

Bouzebda,

Nezzal

2024

MATH

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<abstract><p>$ U $-statistics represent a fundamental class of statistics arising from modeling quantities of interest defined by multi-subject responses. $ U $-statistics generalize the empirical mean of a random variable $ X $ to sums over every $ m $-tuple of distinct observations of $ X $. Stute [182] introduced a class of so-called conditional $ U $-statistics, which may be viewed as a generalization of the Nadaraya-Watson estimates of a regression function. Stute proved their strong pointwise consistency to: $ r^{(m)}(\varphi, \mathbf{t}): = \mathbb{E}[\varphi(Y_{1}, \ldots, Y_{m})|(X_{1}, \ldots, X_{m}) = \mathbf{t}], \; \mbox{for}\; \mathbf{ t}\in \mathcal{X}^{m}. $ In this paper, we are mainly interested in the study of the $ k $NN conditional $ U $-processes in a functional mixing data framework. More precisely, we investigate the weak convergence of the conditional empirical process indexed by a suitable class of functions and of the $ k $NN conditional $ U $-processes when the explicative variable is functional. We treat the uniform central limit theorem in both cases when the class of functions is bounded or unbounded satisfying some moment conditions. The second main contribution of this study is the establishment of a sharp almost complete Uniform consistency in the Number of Neighbors of the constructed estimator. Such a result allows the number of neighbors to vary within a complete range for which the estimator is consistent. Consequently, it represents an interesting guideline in practice to select the optimal bandwidth in nonparametric functional data analysis. These results are proved under some standard structural conditions on the Vapnik-Chervonenkis classes of functions and some mild conditions on the model. The theoretical results established in this paper are (or will be) key tools for further functional data analysis developments. Potential applications include the set indexed conditional <italic>U</italic>-statistics, Kendall rank correlation coefficient, the discrimination problems and the time series prediction from a continuous set of past values.</p></abstract>

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

Cited by 4 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 neighbors consistency and weak convergence of $ k $NN empirical conditional processes and $ k $NN conditional $ U $-processes involving functional mixing data

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