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Wavelet Density and Regression Estimators for Functional Stationary and Ergodic Data: Discrete Time

Author

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  • 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.)

  • Ahoud AL HARBY

    (Department of Mathematics, 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 Compiègne, 60200 Compiègne, France
    These authors contributed equally to this work.)

Abstract

The nonparametric estimation of density and regression function based on functional stationary processes using wavelet bases for Hilbert spaces of functions is investigated in this paper. The mean integrated square error over adapted decomposition spaces is given. To obtain the asymptotic properties of wavelet density and regression estimators, the Martingale method is used. These results are obtained under some mild conditions on the model; aside from ergodicity, no other assumptions are imposed on the data. This paper extends the scope of some previous results for wavelet density and regression estimators by relaxing the independence or the mixing condition to the ergodicity. Potential applications include the conditional distribution, curve discrimination, and time series prediction from a continuous set of past values.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:19:p:3433-:d:921038
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    References listed on IDEAS

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    1. Ibrahim M. Almanjahie & Zoulikha Kaid & Ali Laksaci & Mustapha Rachdi, 2022. "Estimating the Conditional Density in Scalar-On-Function Regression Structure: k -N-N Local Linear Approach," Mathematics, MDPI, vol. 10(6), pages 1-16, March.
    2. Han Shang, 2014. "A survey of functional principal component analysis," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 98(2), pages 121-142, April.
    3. Ibrahim M. Almanjahie & Salim Bouzebda & Zoulikha Kaid & Ali Laksaci, 2022. "Nonparametric estimation of expectile regression in functional dependent data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 34(1), pages 250-281, January.
    4. Vieu, Philippe, 2018. "On dimension reduction models for functional data," Statistics & Probability Letters, Elsevier, vol. 136(C), pages 134-138.
    5. 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.
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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 & 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.

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