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Wavelet Estimation of Partial Derivatives in Multivariate Regression Under Discrete-Time Stationary Ergodic Processes

Author

Listed:
  • Sultana Didi

    (Department of Statistics and Operations Research, College of Sciences, Qassim University, P.O. Box 6688, Buraydah 51452, Saudi Arabia
    These authors contributed equally to this work.)

  • Salim Bouzebda

    (Université de Technologie de Compiègne, LMAC (Laboratory of Applied Mathematics of Compiègne), CS 60 319-60 203 Compiègne, France
    These authors contributed equally to this work.)

Abstract

This study introduces a wavelet-based framework for estimating derivatives of a general regression function within discrete-time, stationary ergodic processes. The analysis focuses on deriving the integrated mean squared error (IMSE) over compact subsets of R d , while also establishing rates of uniform convergence and the asymptotic normality of the proposed estimators. To investigate their asymptotic behavior, we adopt a martingale-based approach specifically adapted to the ergodic nature of the data-generating process. Importantly, the framework imposes no structural assumptions beyond ergodicity, thereby circumventing restrictive dependence conditions. By establishing the limiting behavior of the wavelet estimators under these minimal assumptions, the results extend existing findings for independent data and highlight the flexibility of wavelet methods in more general stochastic settings.

Suggested Citation

  • Sultana Didi & Salim Bouzebda, 2025. "Wavelet Estimation of Partial Derivatives in Multivariate Regression Under Discrete-Time Stationary Ergodic Processes," Mathematics, MDPI, vol. 13(10), pages 1-36, May.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:10:p:1587-:d:1654016
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    References listed on IDEAS

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    1. Bouzebda, Salim & Limnios, Nikolaos, 2013. "On general bootstrap of empirical estimator of a semi-Markov kernel with applications," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 52-62.
    2. F. Roueff & M. S. Taqqu, 2009. "Asymptotic normality of wavelet estimators of the memory parameter for linear processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 30(5), pages 534-558, September.
    3. Christopher R. Genovese & Marco Perone-Pacifico & Isabella Verdinelli & Larry Wasserman, 2016. "Non-parametric inference for density modes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(1), pages 99-126, January.
    4. Tang, Xufei & Xi, Mengmei & Wu, Yi & Wang, Xuejun, 2018. "Asymptotic normality of a wavelet estimator for asymptotically negatively associated errors," Statistics & Probability Letters, Elsevier, vol. 140(C), pages 191-201.
    5. 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.
    6. Salim Bouzebda & Sultana Didi, 2017. "Multivariate wavelet density and regression estimators for stationary and ergodic discrete time processes: Asymptotic results," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 46(3), pages 1367-1406, February.
    7. Dominique Guegan & Sophie A. Ladoucette, 2001. "Non-mixing properties of long memory processes," Post-Print halshs-00193651, HAL.
    8. Evarist Giné & David M. Mason, 2008. "Uniform in Bandwidth Estimation of Integral Functionals of the Density Function," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(4), pages 739-761, December.
    9. Li, Yongming & Yang, Shanchao & Zhou, Yong, 2008. "Consistency and uniformly asymptotic normality of wavelet estimator in regression model with associated samples," Statistics & Probability Letters, Elsevier, vol. 78(17), pages 2947-2956, December.
    10. Paul Deheuvels & David Mason, 2004. "General Asymptotic Confidence Bands Based on Kernel-type Function Estimators," Statistical Inference for Stochastic Processes, Springer, vol. 7(3), pages 225-277, October.
    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. Masry, Elias, 1997. "Multivariate probability density estimation by wavelet methods: Strong consistency and rates for stationary time series," Stochastic Processes and their Applications, Elsevier, vol. 67(2), pages 177-193, May.
    13. Soumaya Allaoui & Salim Bouzebda & Jicheng Liu, 2023. "Multivariate wavelet estimators for weakly dependent processes: strong consistency rate," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 52(23), pages 8317-8350, December.
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