Adaptive estimation for bifurcating Markov chains

S. Valère Bitseki Penda; Marc Hoffmann; Adélaïde Olivier

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arXiv · 1509.03119

Adaptive estimation for bifurcating Markov chains

S. Valère Bitseki Penda·Marc Hoffmann·Adélaïde Olivier

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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.

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BibTeX RIS

S. Valère Bitseki Penda, Marc Hoffmann, Adélaïde Olivier. 2015-09-10. Adaptive estimation for bifurcating Markov chains. https://arxiv.org/abs/1509.03119

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