Threshold Modeling for Bifurcating Autoregression and Large Sample Estimation

Abstract

This article is concerned with threshold modeling of the bifurcating au-toregressive model (BAR) originally suggested by Cowan and Staudte (1986)for tree structured data of cell lineage study where each individual (Xt)gives rise to two o-spring (X2t;X2t+1 ) in the next generation. The triplet(Xt;X2t;X2t+1 ) refers to mother-daughter relationship. In this paper wepropose a threshold model incorporating the dierence of \fertility" of themother for the rst and second o-springs, and thereby extending BAR tothreshold-BAR (TBAR, for short). We derive a sucient condition of sta-tionarity for the suggested TBAR model. Also various inferential methodssuch as least squares (LS), maximum likelihood (ML) and quasi-likelihood(QL) methods are discussed and relevant limiting distributions are obtained.AMS 2000 subject classications.Primary 62M10; Secondary .Keywords.Bifurcating model, fertility, least squares, quasi-likelihood, threshold model.1. IntroductionThe bifurcating autoregressive model (BAR) was originally suggested by Cowanand Staudte (1986) for tree structured data of cell lineage study. Beginning withthe starting valueX1, BAR processfXt; t= 1 ;2;: g is dened by the followingequationXt = X [t=2]+ t; jj < 1; t 2; (1.1)Received May 2006; accepted August 2006.yThis work was supported by grant from KRF(2004-015-C00071).1Corresponding author. Department of Statistics, Sookmyung Women's University, Seoul140-742, Korea (e-mail: shwang@sookmyung.ac.kr)