Weighted estimation and tracking for branching processes with immigration

For branching processes with immigration, we propose an approach which allows us to consistently estimate the means m, λ, and the variances σ², b² of the offspring and immigration distributions, respectively. Generally, statistical results for branching processes are established under the well-known trichotomy m<1, m=1, and m>1. For example, no parameters of the immigration distribution can be consistently estimated if m>1. The purpose of the paper is to obtain, through the introduction of a suitable adaptive control, strongly consistent estimators for all the parameters m, λ, σ², and b² without any restriction on the range of m. Central limit theorems and laws of iterated logarithm are also provided