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A Rademacher–Menchov approach for random coefficient bifurcating autoregressive processes

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  • Bercu, Bernard
  • Blandin, Vassili

Abstract

We investigate the asymptotic behavior of the least squares estimator of the unknown parameters of random coefficient bifurcating autoregressive processes. Under suitable assumptions on inherited and environmental effects, we establish the almost sure convergence of our estimates. In addition, we also prove a quadratic strong law and central limit theorems. Our approach mainly relies on asymptotic results for vector-valued martingales together with the well-known Rademacher–Menchov theorem.

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  • Bercu, Bernard & Blandin, Vassili, 2015. "A Rademacher–Menchov approach for random coefficient bifurcating autoregressive processes," Stochastic Processes and their Applications, Elsevier, vol. 125(4), pages 1218-1243.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:4:p:1218-1243
    DOI: 10.1016/j.spa.2014.10.006
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    References listed on IDEAS

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    1. J. Zhou & I. V. Basawa, 2005. "Maximum Likelihood Estimation for a First‐Order Bifurcating Autoregressive Process with Exponential Errors," Journal of Time Series Analysis, Wiley Blackwell, vol. 26(6), pages 825-842, November.
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    3. Alexander Aue & Lajos Horváth & Josef Steinebach, 2006. "Estimation in Random Coefficient Autoregressive Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 27(1), pages 61-76, January.
    4. de Saporta, Benoîte & Gégout-Petit, Anne & Marsalle, Laurence, 2014. "Statistical study of asymmetry in cell lineage data," Computational Statistics & Data Analysis, Elsevier, vol. 69(C), pages 15-39.
    5. de Saporta, Benoîte & Gégout-Petit, Anne & Marsalle, Laurence, 2012. "Asymmetry tests for bifurcating auto-regressive processes with missing data," Statistics & Probability Letters, Elsevier, vol. 82(7), pages 1439-1444.
    6. Zhou, J. & Basawa, I.V., 2005. "Least-squares estimation for bifurcating autoregressive processes," Statistics & Probability Letters, Elsevier, vol. 74(1), pages 77-88, August.
    7. Nicholls, D. F. & Quinn, B. G., 1981. "The estimation of multivariate random coefficient autoregressive models," Journal of Multivariate Analysis, Elsevier, vol. 11(4), pages 544-555, December.
    8. 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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    Cited by:

    1. 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.
    2. 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.
    3. 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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