Autoregressive Functions Estimation in Nonlinear Bifurcating Autoregressive Models

TL;DR

This paper develops nonparametric estimators for autoregressive functions in bifurcating autoregressive models, analyzing their convergence, asymptotic behavior, and testing for asymmetry in a binary tree structure.

Contribution

It introduces nonparametric Nadaraya-Watson estimators for bifurcating autoregressive models and establishes their convergence, asymptotic normality, and a test for asymmetry.

Findings

Estimators achieve the rate |Tn|^(-β/(2β+1)) in quadratic loss.

Almost sure convergence and asymptotic normality are proven.

An asymptotic test for equality of autoregressive functions is developed.

Abstract

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| --$\beta$/(2$\beta$+1) in quadratic loss over H{\"o}lder 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.