Fixed precision estimator of the offspring mean in branching processes

For the problem of estimating the offspring mean m of a branching process with immigration, we propose a modification of the sequential estimator of m considered in Sriram et al. (1991, Ann. Statist.) and study its nonasymptotic and asymptotic properties. In the nonasymptotic setting, it is shown that the modified estimator is unbiased and has bounded mean squared error (MSE) for all m>0, while the estimator in Sriram et al. is biased and a theoretical bound for its MSE is difficult to obtain. The above result is established for the cases of known and unknown offspring variances, separately. In the asymptotic setting, for the case of m∈(0,1], it is shown that the modified sequential estimator is as efficient as the sequential estimator in Sriram et al. The theoretical results are supported through simulations. Finally, asymptotic normality of the stopping time, for the case of known offspring variance, is also established.