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Nonparametric Recursive Estimation for Multivariate Derivative Functions by Stochastic Approximation Method

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  • Salim Bouzebda

    (LMAC, Université de Technologie de Compiègne)

  • Yousri Slaoui

    (University de Poitiers)

Abstract

Important information concerning a multivariate data set, such as modal regions, is contained in the derivatives of the probability density or regression functions. Despite this importance, nonparametric estimation of higher order derivatives of the density or regression functions have received only relatively scant attention. The main purpose of the present work is to investigate general recursive kernel type estimators of function derivatives. We establish the central limit theorem for the proposed estimators. We discuss the optimal choice of the bandwidth by using the plug in methods. We obtain also the pointwise MDP of these estimators. Finally, we investigate the performance of the methodology for small samples through a short simulation study.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:sankha:v:85:y:2023:i:1:d:10.1007_s13171-021-00272-1
    DOI: 10.1007/s13171-021-00272-1
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    References listed on IDEAS

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    1. Yousri Slaoui, 2014. "Bandwidth Selection for Recursive Kernel Density Estimators Defined by Stochastic Approximation Method," Journal of Probability and Statistics, Hindawi, vol. 2014, pages 1-11, June.
    2. Soumaya Allaoui & Salim Bouzebda & Christophe Chesneau & Jicheng Liu, 2021. "Uniform almost sure convergence and asymptotic distribution of the wavelet-based estimators of partial derivatives of multivariate density function under weak dependence," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 33(2), pages 170-196, April.
    3. Delaigle, A. & Gijbels, I., 2004. "Practical bandwidth selection in deconvolution kernel density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 45(2), pages 249-267, March.
    4. Slaoui, Yousri, 2019. "Wild bootstrap bandwidth selection of recursive nonparametric relative regression for independent functional data," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 494-511.
    5. Puhalskii, A., 1994. "The method of stochastic exponentials for large deviations," Stochastic Processes and their Applications, Elsevier, vol. 54(1), pages 45-70, November.
    6. Yousri Slaoui, 2015. "Plug-in bandwidth selector for recursive kernel regression estimators defined by stochastic approximation method," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 69(4), pages 483-509, November.
    7. Yousri Slaoui, 2020. "Recursive nonparametric regression estimation for dependent strong mixing functional data," Statistical Inference for Stochastic Processes, Springer, vol. 23(3), pages 665-697, October.
    8. Uwe Einmahl & David M. Mason, 2000. "An Empirical Process Approach to the Uniform Consistency of Kernel-Type Function Estimators," Journal of Theoretical Probability, Springer, vol. 13(1), pages 1-37, January.
    9. Bouzebda, Salim & Elhattab, Issam & Seck, Cheikh Tidiane, 2018. "Uniform in bandwidth consistency of nonparametric regression based on copula representation," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 173-182.
    10. 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.
    11. Park, Cheolwoo & Kang, Kee-Hoon, 2008. "SiZer analysis for the comparison of regression curves," Computational Statistics & Data Analysis, Elsevier, vol. 52(8), pages 3954-3970, April.
    12. Salim Bouzebda & Boutheina Nemouchi, 2020. "Uniform consistency and uniform in bandwidth consistency for nonparametric regression estimates and conditional U-statistics involving functional data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 32(2), pages 452-509, April.
    13. 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.
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