Autoregressive Functions Estimation in Nonlinear Bifurcating Autoregressive Models

Siméon Valère Bitseki Penda (1) , Adélaïde Olivier (2, 3, 4)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 MAMBA - Modelling and Analysis for Medical and Biological Applications
3 CREST - Centre de Recherche en Économie et Statistique
4 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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.

Mots clés: asymptotic normality; moderate deviations principle; Bifurcating Markov chains; binary trees; bifurcating autoregressive processes; non-parametric estimation; Nadaraya-Watson estimator; minimax rates of convergence; asymmetry test Mathematics Subject Classification (2010): 62G05; 62G10; 62G20; 60J80; 60F05; 60F10; 60J20; 92D25

Domaines: Théorie [stat.TH]

Siméon Valère Bitseki Penda, Adélaïde Olivier. Autoregressive Functions Estimation in Nonlinear Bifurcating Autoregressive Models. 2015. ⟨hal-01159255v1⟩