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Non parametric estimation for random walks in random environment

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  • Diel, Roland
  • Lerasle, Matthieu

Abstract

We investigate the problem of estimating the cumulative distribution function (c.d.f.) F of a distribution ν from the observation of one trajectory of the random walk in i.i.d. random environment with distribution ν on Z. We first estimate the moments of ν, then combine these moment estimators to obtain a collection of estimators (F̂nM)M≥1 of F, our final estimator is chosen among this collection by Goldenshluger–Lepski’s method. This estimator is easily computable. We derive convergence rates for this estimator depending on the Hölder regularity of F and on the divergence rate of the walk. Our rate is minimal when the chain realizes a trade-off between a fast exploration of the sites, allowing to get more information and a larger number of visits of each site, allowing a better recovery of the environment itself.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:1:p:132-155
    DOI: 10.1016/j.spa.2017.04.011
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    References listed on IDEAS

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    1. Mnatsakanov, Robert M., 2008. "Hausdorff moment problem: Reconstruction of distributions," Statistics & Probability Letters, Elsevier, vol. 78(12), pages 1612-1618, September.
    2. 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.
    3. Mnatsakanov, Robert M., 2008. "Hausdorff moment problem: Reconstruction of probability density functions," Statistics & Probability Letters, Elsevier, vol. 78(13), pages 1869-1877, September.
    4. 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.
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    Cited by:

    1. 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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