Title: Joint Estimation of Offspring Mean and Offspring Variance of Controlled Branching Process

Author(s): Arpita Inamdar and Mohan Kale
Issue: Volume 78 Series A Part 2 Year 2016
Pages: 248 -- 268
The paper discusses the joint estimation of two important parameters of the offspring distribution namely mean and variance of a controlled branching process or $\phi$ branching process. The estimation of these parameters was separately carried out by Gonzalez et al. (${\it Test}$, ${\bf 13}$(2), 465-479, (2004), ${\it Test}$, ${\bf 14}$(1), 199-213, (2005)). The present article is an attempt to show that, the estimators proposed by these authors are also optimal in the sense of estimating functions ($O_F$ optimality). The joint $O_A$ optimality, that is; joint asymptotic properties of these estimators are also established using martingale limit theory. The joint $O_A$ optimality in special case, a model proposed by Dion and Essebbar (1995) for controlled branching process is also discussed.
AMS (2000) subject classification. 60J80, 62M05.
Keywords and phrases: Estimating equations, Joint asymptotic normality, Martingale convergence theorem, Martingale central limit theorem.