2015/06/05 by S. Valère Bitseki Penda, Siméon Valère Bitseki Penda, A. Olivier +3 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #RNA Research and Splicing #Statistical Methods and Inference #Statistics Theory (math.ST) #math.ST #stat.TH
paper · pdf · doi:10.48550/arxiv.1506.01842
openalex publication_date 2015/06/05 · arxiv created 2016/02/11 · arxiv updated 2016/02/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Bifurcating autoregressive processes, which can be seen as an adaptation of au-toregressive processes for a binary tree structure, have been extensively studied during the last decade in a parametric context. In this work we do not specify any a priori form for the two autoregressive functions and we use nonparametric techniques. We investigate both nonasymp-totic and asymptotic behavior of the Nadaraya-Watson type estimators of the autoregressive functions. We build our estimators observing the process on a finite subtree denoted by Tn, up to the depth n. Estimators achieve the classical rate |Tn| --β/(2β+1) in quadratic loss over Hölder classes of smoothness. We prove almost sure convergence, asymptotic normality giving the bias expression when choosing the optimal bandwidth and a moderate deviations principle. Our proofs rely on specific techniques used to study bifurcating Markov chains. Finally, we address the question of asymmetry and develop an asymptotic test for the equality of the two autoregressive functions.