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On the uniform-in-bandwidth consistency of the general conditional U-statistics based on the copula representation

Author

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  • Salim Bouzebda
  • Issam Elhattab
  • Boutheina Nemouchi

Abstract

Stute [Ann. Probab. 19 (1991) 812–825] introduced a class of estimators called conditional U-statistics of \[ \mathbb{E}(\varphi(Y_{1},\ldots,Y_{m})\mid (X_{1},\ldots,X_{m})=\mathbf{ t}),\quad \mbox{for } \mathbf{ t}\in \mathbb{R}^{m}. \] E(φ(Y1,…,Ym)∣(X1,…,Xm)=t),for t∈Rm. In the present work, we provide a new class of estimators of conditional U-statistics. More precisely, we investigate the conditional U-statistics based on copula representation. We establish the uniform-in-bandwidth consistency for the proposed estimator. Our theorems allow data-driven local bandwidths for these statistics. The theoretical uniform consistency results, established in this paper, are (or will be) key tools for many further developments in copula regression analysis. The performance of these procedures is evaluated through simulation in the context of the conditional Kendall's tau.

Suggested Citation

  • Salim Bouzebda & Issam Elhattab & Boutheina Nemouchi, 2021. "On the uniform-in-bandwidth consistency of the general conditional U-statistics based on the copula representation," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 33(2), pages 321-358, April.
    • Handle: RePEc:taf:gnstxx:v:33:y:2021:i:2:p:321-358
    DOI: 10.1080/10485252.2021.1937621
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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. Salim Bouzebda & Inass Soukarieh, 2022. "Non-Parametric Conditional U -Processes for Locally Stationary Functional Random Fields under Stochastic Sampling Design," Mathematics, MDPI, vol. 11(1), pages 1-69, December.
    5. Bouzebda, Salim & Slaoui, Yousri, 2022. "Nonparametric recursive method for moment generating function kernel-type estimators," Statistics & Probability Letters, Elsevier, vol. 184(C).
    6. 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).
    7. 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.
    8. Salim Bouzebda & Boutheina Nemouchi, 2023. "Weak-convergence of empirical conditional processes and conditional U-processes involving functional mixing data," Statistical Inference for Stochastic Processes, Springer, vol. 26(1), pages 33-88, April.

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