## Bernoulli

• Bernoulli
• Volume 23, Number 4B (2017), 3598-3637.

### 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 $1\leq p<\infty$. 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.

#### Article information

Source
Bernoulli, Volume 23, Number 4B (2017), 3598-3637.

Dates
Received: October 2015
Revised: April 2016
First available in Project Euclid: 23 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.bj/1495505103

Digital Object Identifier
doi:10.3150/16-BEJ859

Mathematical Reviews number (MathSciNet)
MR3654817

Zentralblatt MATH identifier
06778297

#### Citation

Bitseki Penda, S. Valère; Hoffmann, Marc; Olivier, Adélaïde. Adaptive estimation for bifurcating Markov chains. Bernoulli 23 (2017), no. 4B, 3598--3637. doi:10.3150/16-BEJ859. https://projecteuclid.org/euclid.bj/1495505103

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