Estimation of the Means in the Branching Process with Immigration

Abstract

Let $\{X_n\}$ be the branching process with immigration and let $m$ and $\lambda$ be the means of the offspring and immigration distributions, respectively. Estimation results for $m$ and $\lambda$ were obtained in the literature for the subcritical $(m < 1)$ and supercritical $(m > 1)$ cases, but no unified estimation procedure was developed, which would allow inference without knowledge of the range of $m$. The goal of this paper is to investigate this problem.