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Moderate deviation principles for kernel estimator of invariant density in bifurcating Markov chains

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  • Bitseki Penda, S. Valère

Abstract

Bitseki and Delmas (2022) have studied recently the central limit theorem for kernel estimator of invariant density in bifurcating Markov chains. We complete their work by proving a moderate deviation principle for this estimator. Unlike the work of Bitseki and Gorgui (2022), it is interesting to see that the distinction of the two regimes disappears and that we are able to get moderate deviation principle for large values of the ergodic rate. It is also interesting and surprising to see that for moderate deviation principle, the ergodic rate begins to have an impact on the choice of the bandwidth for values smaller than in the context of central limit theorem studied by Bitseki and Delmas (2022).

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:158:y:2023:i:c:p:282-314
    DOI: 10.1016/j.spa.2023.01.004
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    References listed on IDEAS

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    1. S. Valère Bitseki Penda & Angelina Roche, 2020. "Local bandwidth selection for kernel density estimation in a bifurcating Markov chain model," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 32(3), pages 535-562, July.
    2. George Roussas, 1969. "Nonparametric estimation in Markov processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 21(1), pages 73-87, December.
    3. 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.
    4. Fuqing Gao, 2003. "Moderate Deviations and Large Deviations for Kernel Density Estimators," Journal of Theoretical Probability, Springer, vol. 16(2), pages 401-418, April.
    5. 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.
    6. 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.
    7. Delmas, Jean-François & Marsalle, Laurence, 2010. "Detection of cellular aging in a Galton-Watson process," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2495-2519, December.
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