On sequential estimation for branching processes with immigration

Consider a Galton–Watson process with immigration. The limiting distributions of the nonsequential estimators of the offspring mean have been proved to be drastically different for the critical case and subcritical and supercritical cases. A sequential estimator, proposed by Sriram et al. (Ann. Statist. 19 (1991) 2232), was shown to be asymptotically normal for both the subcritical and critical cases. Based on a certain stopping rule, we construct a class of two-stage estimators for the offspring mean. These estimators are shown to be asymptotically normal for all the three cases. This gives, without assuming any prior knowledge, a unified estimation and inference procedure for the offspring mean.