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Autoregressive functions estimation in nonlinear bifurcating autoregressive models

Author

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  • S. Valère Bitseki Penda

    (Université Bourgogne Franche-Comté, CNRS, UMR [5584], IMB)

  • Adélaïde Olivier

    (Université Paris-Dauphine, PSL Research University, CNRS, UMR [7534], CEREMADE)

Abstract

Bifurcating autoregressive processes, which can be seen as an adaptation of autoregressive processes for a binary tree structure, have been extensively studied during the last decade in a parametric context. In this work we do not specify any a priori form for the two autoregressive functions and we use nonparametric techniques. We investigate both nonasymptotic and asymptotic behaviour of the Nadaraya–Watson type estimators of the autoregressive functions. We build our estimators observing the process on a finite subtree denoted by $$\mathbb {T}_n$$ T n , up to the depth n. Estimators achieve the classical rate $$|\mathbb {T}_n|^{-\beta /(2\beta +1)}$$ | T n | - β / ( 2 β + 1 ) in quadratic loss over Hölder classes of smoothness. We prove almost sure convergence, asymptotic normality giving the bias expression when choosing the optimal bandwidth. Finally, we address the question of asymmetry: we develop an asymptotic test for the equality of the two autoregressive functions which we implement both on simulated and real data.

Suggested Citation

  • S. Valère Bitseki Penda & Adélaïde Olivier, 2017. "Autoregressive functions estimation in nonlinear bifurcating autoregressive models," Statistical Inference for Stochastic Processes, Springer, vol. 20(2), pages 179-210, July.
  • Handle: RePEc:spr:sistpr:v:20:y:2017:i:2:d:10.1007_s11203-016-9140-6
    DOI: 10.1007/s11203-016-9140-6
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    References listed on IDEAS

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    Cited by:

    1. Bitseki Penda, S. Valère, 2023. "Moderate deviation principles for kernel estimator of invariant density in bifurcating Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 282-314.
    2. Hoffmann, Marc & Marguet, Aline, 2019. "Statistical estimation in a randomly structured branching population," Stochastic Processes and their Applications, Elsevier, vol. 129(12), pages 5236-5277.
    3. Bitseki Penda, S. Valère & Olivier, Adélaïde, 2018. "Moderate deviation principle in nonlinear bifurcating autoregressive models," Statistics & Probability Letters, Elsevier, vol. 138(C), pages 20-26.
    4. Vincent Bansaye & S. Valère Bitseki Penda, 2021. "A Phase Transition for Large Values of Bifurcating Autoregressive Models," Journal of Theoretical Probability, Springer, vol. 34(4), pages 2081-2116, December.

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