Adaptive estimation for bifurcating Markov chains

TL;DR

This paper develops Bernstein-type deviation inequalities for bifurcating Markov chains, enabling the construction of adaptive nonparametric estimators for transition densities, with applications to growth-fragmentation models and bifurcating autoregressive processes.

Contribution

It introduces Bernstein-type deviation inequalities for BMCs and applies them to create nearly minimax optimal nonparametric estimators for various densities.

Findings

Established Bernstein-type deviation inequalities for BMCs.

Constructed adaptive wavelet thresholding estimators for densities.

Applied results to growth-fragmentation models and autoregressive processes.

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.