Maximum likelihood estimation and the local asymptotic mixed normality in a second‐order branching process with continuous state space

Abstract

A second‐order branching process with continuous state space (SOBPCS) is considered. Maximum likelihood (ML) estimation in SOBPCS is studied under the assumption that the underlying subordinators are gamma processes. This estimation is discussed under two sampling schemes. In the first scheme, we assume that all generation sizes as well as the contributions of lag 1 and lag 2 in each generation are observable. In the second scheme, we assume that only generation sizes are observable. The derived estimators under each scheme are shown to be consistent and asymptotically normal (CAN). Also, it is shown that the SOBPCS belongs to the local asymptotic mixed normal (LAMN) family of models. A simulation study is carried out to verify the results. We apply our results to the optical density data emerging through growth of Bacilllus subtilis bacteria.