Autoregressive functions estimation in nonlinear bifurcating autoregressive models

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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 \(\mathbb {T}_n\), up to the depth n. Estimators achieve the classical rate \(|\mathbb {T}_n|^{-\beta /(2\beta +1)}\) 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 

Mathematics Subject Classification

62G05 62G10 62G20 60J80 60F05 60J20 92D25 

Notes

Acknowledgments

We are grateful to A. Guillin and M. Hoffmann for helpful comments and G. Fort for useful discussions. We thank E. Löcherbach and P. Reynaud-Bouret for a careful reading. We also deeply thank N. Krell for discussions on real data and E. Stewart for sharing his data. S.V. B.P. thanks the Hadamard Mathematics Labex of the Fondation Mathématique Jacques Hadamard for financial support.

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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • S. Valère Bitseki Penda
    • 1
  • Adélaïde Olivier
    • 2
  1. 1.Université Bourgogne Franche-Comté, CNRS, UMR [5584], IMBDijonFrance
  2. 2.Université Paris-Dauphine, PSL Research University, CNRS, UMR [7534], CEREMADEParisFrance

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