Open Access
November 2017 Adaptive estimation for bifurcating Markov chains
S. Valère Bitseki Penda, Marc Hoffmann, Adélaïde Olivier
Bernoulli 23(4B): 3598-3637 (November 2017). DOI: 10.3150/16-BEJ859


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 $1\leq p<\infty$. 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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S. Valère Bitseki Penda. Marc Hoffmann. Adélaïde Olivier. "Adaptive estimation for bifurcating Markov chains." Bernoulli 23 (4B) 3598 - 3637, November 2017.


Received: 1 October 2015; Revised: 1 April 2016; Published: November 2017
First available in Project Euclid: 23 May 2017

zbMATH: 06778297
MathSciNet: MR3654817
Digital Object Identifier: 10.3150/16-BEJ859

Keywords: Bifurcating autoregressive process , Bifurcating Markov chains , binary trees , Deviations inequalities , growth-fragmentation processes , minimax rates of convergence , Nonparametric adaptive estimation

Rights: Copyright © 2017 Bernoulli Society for Mathematical Statistics and Probability

Vol.23 • No. 4B • November 2017
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