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Estimation of the offspring mean in a controlled branching process with a random control function. (English) Zbl 1114.62082
Summary: Controlled branching processes (CBP) with a random control function provide a useful way to model generation sizes in population dynamics studies, where control on the growth of the population size is necessary at each generation. An important special case of this process is the well known branching process with immigration. Motivated by work of C. Z. Wei and J. Winnicki [Estimation of the mean in the branching process with immigration. Ann. Stat. 18, No. 4, 1757–1773 (1990; Zbl 0736.62071)], we develop a weighted conditional least squares estimator of the offspring mean of the CBP and derive the asymptotic limit distribution of the estimator when the process is subcritical, critical and supercritical. Moreover, we show the strong consistency of this estimator in all the cases. The results obtained here extend those of Wei and Winnicki for branching processes with immigration and provide a unified limit theory of estimation.

##### MSC:
 62M05 Markov processes: estimation; hidden Markov models 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60F05 Central limit and other weak theorems 62E20 Asymptotic distribution theory in statistics
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