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Exponential Bounds and Convergence Rates of Sieve Estimators for Functional Autoregressive Processes

Author

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  • Nesrine Kara Terki

    (Higher School of Management-Tlemcen)

  • Tahar Mourid

    (University of Abou Bakr Belkaid Tlemcen)

Abstract

In the following study, we deal with the exponential bounds and rates for a class of sieve estimators of Grenander for Functional Autoregressive Processes when the parameter operator belongs to the parameter space of Hilbert-Schmidt operators. Two classes of parameter operators are considered where we state clearly sieve estimators formulas and derive corresponding exponential bounds. These results are applied to establish their almost sure convergence and almost complete convergence. Then, we determine rates of convergence of sieve estimators in each class. The numerical studies illustrate the performance of the sieve predictors and give comparisons with other existing prediction methods both on simulated and real functional data sets exhibiting competitive results.

Suggested Citation

  • Nesrine Kara Terki & Tahar Mourid, 2024. "Exponential Bounds and Convergence Rates of Sieve Estimators for Functional Autoregressive Processes," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 86(1), pages 364-391, February.
    • Handle: RePEc:spr:sankha:v:86:y:2024:i:1:d:10.1007_s13171-023-00322-w
    DOI: 10.1007/s13171-023-00322-w
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    References listed on IDEAS

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    1. Kara-Terki, Nesrine & Mourid, Tahar, 2016. "On local asymptotic normality for functional autoregressive processes," Journal of Multivariate Analysis, Elsevier, vol. 148(C), pages 120-140.
    2. Berhoune, Kamila & Bensmain, Nawel, 2018. "Sieves estimator of functional autoregressive process," Statistics & Probability Letters, Elsevier, vol. 135(C), pages 60-69.
    3. Mourid, Tahar & Bensmain, Nawel, 2006. "Sieves estimator of the operator of a functional autoregressive process," Statistics & Probability Letters, Elsevier, vol. 76(1), pages 93-108, January.
    4. Philippe C. Besse & Herve Cardot & David B. Stephenson, 2000. "Autoregressive Forecasting of Some Functional Climatic Variations," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 27(4), pages 673-687, December.
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