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On quasi-likelihood estimation for branching processes with immigration. (English. French summary) Zbl 1349.62388

Summary: 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 in 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 be collected over a long period of time.

MSC:

62M09 Non-Markovian processes: estimation
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
62F10 Point estimation
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