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Uniform convergence rate of the kernel regression estimator adaptive to intrinsic dimension in presence of censored data

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  • Salim Bouzebda
  • Thouria El-hadjali

Abstract

The focus of the present paper is on the uniform in bandwidth consistency of kernel-type estimators of the regression function $\mathbb {E}(\Psi (\mathbf {Y})\mid \mathbf {{ X}}=\mathbf { x}) $E(Ψ(Y)∣X=x) derived by modern empirical process theory, under weaker conditions on the kernel than previously used in the literature. Our theorems allow data-driven local bandwidths for these statistics. We extend existing uniform bounds on kernel regression estimator and making it adaptive to the intrinsic dimension of the underlying distribution of $\mathbf {X} $X which will be characterising by the so-called intrinsic dimension. Moreover, we show, in the same context, the uniform in bandwidth consistency for nonparametric inverse probability of censoring weighted (I.P.C.W.) estimators of the regression function under random censorship. Statistical applications to the kernel-type estimators (density, regression, conditional distribution, derivative functions, entropy, mode and additive models) are given to motivate these results.

Suggested Citation

  • Salim Bouzebda & Thouria El-hadjali, 2020. "Uniform convergence rate of the kernel regression estimator adaptive to intrinsic dimension in presence of censored data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 32(4), pages 864-914, October.
  • Handle: RePEc:taf:gnstxx:v:32:y:2020:i:4:p:864-914
    DOI: 10.1080/10485252.2020.1834107
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    Cited by:

    1. Salim Bouzebda & Amel Nezzal & Tarek Zari, 2022. "Uniform Consistency for Functional Conditional U -Statistics Using Delta-Sequences," Mathematics, MDPI, vol. 11(1), pages 1-39, December.
    2. Sultana Didi & Salim Bouzebda, 2022. "Wavelet Density and Regression Estimators for Continuous Time Functional Stationary and Ergodic Processes," Mathematics, MDPI, vol. 10(22), pages 1-37, November.
    3. Salim Bouzebda & Thouria El-hadjali & Anouar Abdeldjaoued Ferfache, 2023. "Uniform in Bandwidth Consistency of Conditional U-statistics Adaptive to Intrinsic Dimension in Presence of Censored Data," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(2), pages 1548-1606, August.
    4. Bouzebda, Salim & Slaoui, Yousri, 2022. "Nonparametric recursive method for moment generating function kernel-type estimators," Statistics & Probability Letters, Elsevier, vol. 184(C).
    5. Soukarieh, Inass & Bouzebda, Salim, 2023. "Renewal type bootstrap for increasing degree U-process of a Markov chain," Journal of Multivariate Analysis, Elsevier, vol. 195(C).
    6. Salim Bouzebda & Yousri Slaoui, 2023. "Nonparametric Recursive Estimation for Multivariate Derivative Functions by Stochastic Approximation Method," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(1), pages 658-690, February.

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