Maximum likelihood estimation and expectation–maximization algorithm for controlled branching processes

The controlled branching process is a generalization of the classical Bienaymé–Galton–Watson branching process. It is a useful model for describing the evolution of populations in which the population size at each generation needs to be controlled. The maximum likelihood estimation of the parameters of interest for this process is addressed under various sample schemes. Firstly, assuming that the entire family tree can be observed, the corresponding estimators are obtained and their asymptotic properties investigated. Secondly, since in practice it is not usual to observe such a sample, the maximum likelihood estimation is initially considered using the sample given by the total number of individuals and progenitors of each generation, and then using the sample given by only the generation sizes. Expectation–maximization algorithms are developed to address these problems as incomplete data estimation problems. The accuracy of the procedures is illustrated by means of a simulated example.