Joint estimation of the offspring mean and offspring variance of a second order branching process

In this paper, under the assumption that the offspring distributions of lag one and lag two are identical, we discuss the joint estimation of the offspring mean and the offspring variance of a second order branching process. The proposed estimator is shown to be finite sample optimal, if the common offspring distribution is mesokurtic and symmetric. The conditional asymptotic joint normality of the proposed estimator is established using the martingale limit theory.