Adaptive estimation for bifurcating Markov chains
Hoffmann, Marc; Olivier, Adélaïde; Valère Bitseki Penda, Siméon (2017), Adaptive estimation for bifurcating Markov chains, Bernoulli, 23, 4B, p. 3598 - 3637. 10.3150/16-BEJ859
TypeArticle accepté pour publication ou publié
International Statistical Institute
3598 - 3637
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Abstract (EN)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.
Subjects / KeywordsBinary trees; Bifurcating Markov chains; Nonparametric adaptive estimation; Bifurcating autoregressive process; Growth-fragmentation processes; Minimax rates of convergence; Deviations inequalities
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