Electronic Journal of Statistics ISSN: 1935-7524 Parameters estimation for asymmetric bifurcating autoregressive processes with missing data Benoîte de Saporta Université de Bordeaux, GREThA CNRS UMR 5113, IMB CNRS UMR 5251 and INRIA Bordeaux Sud Ouest team CQFD, France e-mail:
[email protected] and Anne Gégout-Petit Université de Bordeaux, IMB, CNRS UMR 525 and INRIA Bordeaux Sud Ouest team CQFD, France e-mail:
[email protected] and Laurence Marsalle Université de Lille 1, Laboratoire Paul Painlevé, CNRS UMR 8524, France e-mail:
[email protected] Abstract: We estimate the unknown parameters of an asymmetric bi- furcating autoregressive process (BAR) when some of the data are missing. In this aim, we model the observed data by a two-type Galton-Watson pro- cess consistent with the binary Bee structure of the data. Under indepen- dence between the process leading to the missing data and the BAR process and suitable assumptions on the driven noise, we establish the strong con- sistency of our estimators on the set of non-extinction of the Galton-Watson process, via a martingale approach. We also prove a quadratic strong law and the asymptotic normality. AMS 2000 subject classifications: Primary 62F12, 62M09; secondary 60J80, 92D25, 60G42. Keywords and phrases: least squares estimation, bifurcating autoregres- sive process, missing data, Galton-Watson process, joint model, martin- gales, limit theorems. arXiv:1012.2012v3 [math.PR] 27 Sep 2011 1. Introduction Bifurcating autoregressive processes (BAR) generalize autoregressive (AR) pro- cesses, when the data have a binary tree structure. Typically, they are involved in modeling cell lineage data, since each cell in one generation gives birth to two offspring in the next one.