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Uniform almost sure convergence and asymptotic distribution of the wavelet-based estimators of partial derivatives of multivariate density function under weak dependence

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  • Soumaya Allaoui
  • Salim Bouzebda
  • Christophe Chesneau
  • Jicheng Liu

Abstract

This paper is devoted to the estimation of partial derivatives of multivariate density functions. In this regard, nonparametric linear wavelet-based estimators are introduced, showing their attractive properties from the theoretical point of view. In particular, we prove the strong uniform consistency properties of these estimators, over compact subsets of $ \mathbb {R}^{d} $ Rd, with the determination of the corresponding convergence rates. Then, we establish the asymptotic normality of these estimators. As a main contribution, we relax some standard dependence conditions; our results hold under a weak dependence condition allowing the consideration of mixing, association, Gaussian sequences and Bernoulli shifts.

Suggested Citation

  • 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.
  • Handle: RePEc:taf:gnstxx:v:33:y:2021:i:2:p:170-196
    DOI: 10.1080/10485252.2021.1925668
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    Cited by:

    1. 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.
    2. 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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