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.
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Bernoulli, Bernoulli Society for Mathematical Statistics and Probability, 2017, 23 (4B), pp.3598 - 3637. 〈10.3150/16-BEJ859〉
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Siméon Valère Bitseki Penda, Marc Hoffmann, Adélaïde Olivier. Adaptive estimation for bifurcating Markov chains. Bernoulli, Bernoulli Society for Mathematical Statistics and Probability, 2017, 23 (4B), pp.3598 - 3637. 〈10.3150/16-BEJ859〉. 〈hal-01254200〉

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