Estimators in Branching Processes with Immigration

Abstract

In the present paper we consider the branching process with immigration and its relationship to the Bienayme - Galton - Watson process with a random number of ancestors. Several estimators of the immigration component are considered - the conditional least squares estimator of Heyde - Seneta, the conditional weighted least squares estimator of Wei - Winnicki and the estimator of Dion and Yanev. Their comparison is based on simulations of the entire immigration family trees and computational results. The asymptotic normality of the estimator of Dion and Yanev is combined with the general idea of the trimmed and weighted maximum likelihood. As a result, robust modifications of the immigration component estimator is proposed. They are based on one and several realizations of the entire family tree and are studied via simulations and numerical results.