Autoregressive functions estimation in nonlinear bifurcating autoregressive models

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

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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 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: Bifurcating autoregressive processes; Nonparametric estimation; Nadaraya–Watson estimator; Asymptotic normality; Minimax rates of convergence; Asymmetry test; Bifurcating Markov chains; Binary trees

Domaines: Théorie [stat.TH]

Siméon Valère Bitseki Penda, Adélaïde Olivier. Autoregressive functions estimation in nonlinear bifurcating autoregressive models. Statistical Inference for Stochastic Processes, 2017, 20 (2), pp.179 - 210. ⟨10.1007/s11203-016-9140-6⟩. ⟨hal-01159255v2⟩