Autoregressive Functions Estimation in Nonlinear Bifurcating Autoregressive Models
Valère Bitseki Penda, Siméon; Olivier, Adélaïde (2017), Autoregressive Functions Estimation in Nonlinear Bifurcating Autoregressive Models, Statistical Inference for Stochastic Processes, 20, 2, p. 179–210. 10.1007/s11203-016-9140-6
TypeArticle accepté pour publication ou publié
Journal nameStatistical Inference for Stochastic Processes
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Abstract (EN)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.
Subjects / Keywordsminimax rates of convergence; Bifurcating Markov chains; binary trees; bifurcating autoregressive processes; non-parametric estimation; asymptotic normality; moderate deviations principle; asymmetry test; Nadaraya-Watson estimator
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