Offspring Mean Estimators in Branching Processes with Immigration

2000 Mathematics Subject Classification: 60J80.In the present paper we consider the discrete time branching process with immigration and its relationship to the Bienayme-Galton-Watson process with a random number of ancestors. Several estimators of the offspring mean are considered - the Harris estimator, 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 properties are compared using computational results based on simulations of the entire immigration family trees. 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 offspring mean estimator is proposed.The paper is partially supported by the National Science Fund of Bulgaria, Grant No VU-MI-105/2005