Parameters estimation for asymmetric bifurcating autoregressive processes with missing data

TL;DR

This paper develops consistent estimators for parameters of asymmetric bifurcating autoregressive processes with missing data, using a Galton-Watson model and martingale techniques, and proves their asymptotic properties.

Contribution

It introduces a novel estimation method for BAR processes with missing data, establishing strong consistency, quadratic strong law, and asymptotic normality under new assumptions.

Findings

Establishes strong consistency of estimators.

Proves quadratic strong law for estimators.

Demonstrates asymptotic normality of estimators.

Abstract

We estimate the unknown parameters of an asymmetric bifurcating autoregressive process (BAR) when some of the data are missing. In this aim, we model the observed data by a two-type Galton-Watson process consistent with the binary tree structure of the data. Under independence between the process leading to the missing data and the BAR process and suitable assumptions on the driven noise, we establish the strong consistency of our estimators on the set of non-extinction of the Galton-Watson, via a martingale approach. We also prove a quadratic strong law and the asymptotic normality.