Consistency of Estimators in Partially Observed Subcritical Branching Processes with Immigration

It is known that in subcritical branching process with stationary immigration the average population size for the first n generations and the ratio of the reproduction process to the total progeny are strongly consistent estimators for the mean of the stationary distribution and for the offspring mean, respectively. We prove that the same estimators remain strongly consistent, if we have only partial observations of the population and the number of immigrants. We also show that the rates of convergence of the estimators to the true values of the parameters are the same as in the case of complete observation