Asymptotic behavior of CLS estimators for $2$-type doubly symmetric critical Galton-Watson processes with immigration

TL;DR

This paper investigates the long-term properties of CLS estimators for offspring means and the criticality parameter in a specific 2-type critical Galton-Watson process with immigration, providing insights into their asymptotic behavior.

Contribution

It characterizes the asymptotic distribution of CLS estimators for key parameters in a 2-type critical Galton-Watson process with immigration, a novel analysis for this class of processes.

Findings

Asymptotic distribution of CLS estimators derived

Estimates for offspring means and criticality parameter established

Results applicable to 2-type critical Galton-Watson processes with immigration

Abstract

In this paper, the asymptotic behavior of the conditional least squares (CLS) estimators of the offspring means $(\alpha,\beta)$ and of the criticality parameter $\varrho:=\alpha+\beta$ for a $2$-type critical doubly symmetric positively regular Galton-Watson branching process with immigration is described.