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Wavelet Density and Regression Estimators for Continuous Time Functional Stationary and Ergodic Processes

Author

Listed:
  • Sultana Didi

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

  • Salim Bouzebda

    (LMAC (Laboratory of Applied Mathematics of Compiègne), Université de Technologie de Compiégne, 60200 Compiègne, France
    These authors contributed equally to this work.)

Abstract

In this study, we look at the wavelet basis for the nonparametric estimation of density and regression functions for continuous functional stationary processes in Hilbert space. The mean integrated squared error for a small subset is established. We employ a martingale approach to obtain the asymptotic properties of these wavelet estimators. These findings are established under rather broad assumptions. All we assume about the data is that they are ergodic, but beyond that, we make no assumptions. In this paper, the mean integrated squared error findings in the independence or mixing setting were generalized to the ergodic setting. The theoretical results presented in this study are (or will be) valuable resources for various cutting-edge functional data analysis applications. Applications include conditional distribution, conditional quantile, entropy, and curve discrimination.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:22:p:4356-:d:978020
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    References listed on IDEAS

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    1. Sultana DIDI & Ahoud AL HARBY & Salim BOUZEBDA, 2022. "Wavelet Density and Regression Estimators for Functional Stationary and Ergodic Data: Discrete Time," Mathematics, MDPI, vol. 10(19), pages 1-33, September.
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    4. Litimein, Ouahiba & Laksaci, Ali & Mechab, Boubaker & Bouzebda, Salim, 2023. "Local linear estimate of the functional expectile regression," Statistics & Probability Letters, Elsevier, vol. 192(C).
    5. 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.
    6. Salim Bouzebda & Sultana Didi, 2022. "Some results about kernel estimators for function derivatives based on stationary and ergodic continuous time processes with applications," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 51(12), pages 3886-3933, May.
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    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.

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