Adaptive Estimation for Bifurcating Markov Chains

Abstract : In a first part, we prove Bernstein-type deviation inequalities for bifurcating Markov chains (BMC) under a geometric ergodicity assumption, completing former results of Guyon and Bitseki Penda, Djellout and Guillin. These preliminary results are the key ingredient to implement nonparametric wavelet thresholding estimation procedures: in a second part, we construct nonparametric estimators of the transition density of a BMC, of its mean transition density and of the corresponding invariant density, and show smoothness adaptation over various multivariate Besov classes under L^p-loss error, for 1 ≤ p < ∞. We prove that our estimators are (nearly) optimal in a minimax sense. As an application, we obtain new results for the estimation of the splitting size-dependent rate of growth-fragmentation models and we extend the statistical study of bifurcating autoregressive processes.
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Pré-publication, Document de travail
2016
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https://hal.archives-ouvertes.fr/hal-01254200
Contributeur : Adélaïde Olivier <>
Soumis le : lundi 11 janvier 2016 - 21:11:22
Dernière modification le : mardi 11 octobre 2016 - 15:08:10
Document(s) archivé(s) le : mardi 12 avril 2016 - 11:42:33

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  • HAL Id : hal-01254200, version 1
  • ARXIV : 1509.03119

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Bitseki Penda, Marc Hoffmann, Adélaïde Olivier. Adaptive Estimation for Bifurcating Markov Chains. 2016. <hal-01254200>

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