On quasi-likelihood estimation for branching processes with immigration

In the theory of estimation for branching processes it is well known that, in the super-critical case, the so-called conditional least-squares and the conditional weighted least-squares methods may not yield unbiased and hence consistent estimates for the mean parameters of the offspring and immigration distributions. In this paper, the authors propose a new conditional quasi-likelihood method in the context of negative binomial offspring and immigration distributions that provides mean estimates with smaller mean squared errors M the super-critical case as compared to the previous approaches. Further, they simplify the conditional quasi-likelihood estimating equations both for the mean and the variance parameters under a special model with binary offspring distribution appropriate for a controlled population. It is also demonstrated empirically that a reasonable estimate for the variance or overdispersion parameter requires that the data he collected over a long period of time. The Canadian Journal of Statistics 38: 290-313; 2010 (C) 2010 Statistical Society of Canada