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Maximum likelihood estimator consistency for a ballistic random walk in a parametric random environment

Author

Listed:
  • Comets, Francis
  • Falconnet, Mikael
  • Loukianov, Oleg
  • Loukianova, Dasha
  • Matias, Catherine

Abstract

We consider a one dimensional ballistic random walk evolving in an i.i.d. parametric random environment. We provide a maximum likelihood estimation procedure of the parameters based on a single observation of the path till the time it reaches a distant site, and prove that the estimator is consistent as the distant site tends to infinity. Our main tool consists in using the link between random walks and branching processes in random environments and explicitly characterising the limiting distribution of the process that arises. We also explore the numerical performance of our estimation procedure.

Suggested Citation

  • Comets, Francis & Falconnet, Mikael & Loukianov, Oleg & Loukianova, Dasha & Matias, Catherine, 2014. "Maximum likelihood estimator consistency for a ballistic random walk in a parametric random environment," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 268-288.
    • Handle: RePEc:eee:spapps:v:124:y:2014:i:1:p:268-288
    DOI: 10.1016/j.spa.2013.08.002
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    Cited by:

    1. Grégoire Véchambre, 2023. "Almost Sure Behavior for the Local Time of a Diffusion in a Spectrally Negative Lévy Environment," Journal of Theoretical Probability, Springer, vol. 36(2), pages 876-925, June.
    2. Comets, Francis & Falconnet, Mikael & Loukianov, Oleg & Loukianova, Dasha, 2016. "Maximum likelihood estimator consistency for recurrent random walk in a parametric random environment with finite support," Stochastic Processes and their Applications, Elsevier, vol. 126(11), pages 3578-3604.
    3. Diel, Roland & Lerasle, Matthieu, 2018. "Non parametric estimation for random walks in random environment," Stochastic Processes and their Applications, Elsevier, vol. 128(1), pages 132-155.
    4. Andreoletti, Pierre & Diel, Roland, 2020. "The heavy range of randomly biased walks on trees," Stochastic Processes and their Applications, Elsevier, vol. 130(2), pages 962-999.
    5. Rémillard, Bruno N. & Vaillancourt, Jean, 2019. "Detecting periodicity from the trajectory of a random walk in random environment," Statistics & Probability Letters, Elsevier, vol. 155(C), pages 1-1.

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