Autoregressive functions estimation in nonlinear bifurcating autoregressive models

S. Valère Bitseki Penda () and Adélaïde Olivier ()
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S. Valère Bitseki Penda: Université Bourgogne Franche-Comté, CNRS, UMR [5584], IMB
Adélaïde Olivier: Université Paris-Dauphine, PSL Research University, CNRS, UMR [7534], CEREMADE

Statistical Inference for Stochastic Processes, 2017, vol. 20, issue 2, No 2, 179-210

Abstract: Abstract Bifurcating autoregressive processes, which can be seen as an adaptation of autoregressive 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 nonasymptotic and asymptotic behaviour of the Nadaraya–Watson type estimators of the autoregressive functions. We build our estimators observing the process on a finite subtree denoted by $$\mathbb {T}_n$$ T n , up to the depth n. Estimators achieve the classical rate $$|\mathbb {T}_n|^{-\beta /(2\beta +1)}$$ | T n | - β / ( 2 β + 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. Finally, we address the question of asymmetry: we develop an asymptotic test for the equality of the two autoregressive functions which we implement both on simulated and real data.

Keywords: Bifurcating Markov chains; Binary trees; Bifurcating autoregressive processes; Nonparametric estimation; Nadaraya–Watson estimator; Minimax rates of convergence; Asymptotic normality; Asymmetry test; 62G05; 62G10; 62G20; 60J80; 60F05; 60J20; 92D25 (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (4)

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DOI: 10.1007/s11203-016-9140-6

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