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Bifurcating autoregressive processes, which can be seen as an adaptation ofau-toregressive processes for a binary tree structure, have been extensivelystudied during the last decade in a parametric context. In this work we do notspecify any a priori form for the two autoregressive functions and we usenonparametric techniques. We investigate both nonasymp-totic and asymptoticbehavior of the Nadaraya-Watson type estimators of the autoregressivefunctions. We build our estimators observing the process on a finite subtreedenoted by Tn, up to the depth n. Estimators achieve the classical rate |Tn|--$\beta$/(2$\beta$+1) in quadratic loss over H{\"o}lder classes of smoothness.We prove almost sure convergence, asymptotic normality giving the biasexpression when choosing the optimal bandwidth and a moderate deviationsprinciple. Our proofs rely on specific techniques used to study bifurcatingMarkov chains. Finally, we address the question of asymmetry and develop anasymptotic test for the equality of the two autoregressive functions.

Autoregressive Functions Estimation in Nonlinear Bifurcating Autoregressive Models