Abstract
Let \(\{Z(k), k\geq 0\}\) be a branching stochastic process with non-stationary immigration given by offspring distribution \(\{p_{j}(\theta),j\geq 0\}\) depending on unknown parameter \(\theta\in \Theta\). We estimate θ by an estimator \(\hat{\theta}_{n}\) based on sample \(\mathcal{X}_{n}=\{Z(i), i=1, {\ldots}, n\}\). Given \(\mathcal{X}_{n}\), we generate bootstrap branching process \(\{Z^{\mathcal{X}_{n}}(k), k\geq 0\}\) for each \( n=1, 2, {\ldots}\) with offspring distribution \(\{p_{j}(\hat{\theta}_{n}), j\geq 0\}\). In the paper we address the following question: How good must be estimator \(\hat{\theta}_{n}\), the bootstrap process to have the same asymptotic properties as the original process? We obtain conditions for the estimator which are sufficient and necessary for this in critical case. To derive these conditions we investigate a weighted sum of martingale differences generated by an array of branching processes. We provide a general functional limit theorem for this sum, which includes critical or nearly critical processes with increasing or stationary immigration and with large or fixed number of initial ancestors. It also includes processes without immigration with increasing random number of initial individuals. Possible applications in estimation theory of branching processes are also be provided.
Mathematics Subject Classification (2000): 60J80
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References
Datta, S., Sriram, T.N.: A modified bootstrap for branching processes with immigration. Stoch. Proc. Appl. 56, 275–294 (1995)
Grimvall, A.: On the convergence of sequences of branching processes. Ann. Probab. 2, 1027–1045 (1974)
Lindvall, T.: Convergence of critical Galton–Watson branching processes. J. Appl. Probab. 9, 445–450 (1972)
Rahimov, I.: Functional limit theorems for critical processes with immigration. Adv. Appl. Probab. 39 (4), 1054–1069 (2007)
Rahimov, I.: Limit distributions for weighted estimators of the offspring mean in a branching process. TEST (2009) doi: 10.1007/s11749-008-0124-8
Sweeting, T. J.: On conditional weak convergence. J. Theor. Probab. 2 (2), 461–474 (1989)
Xiong, S., Li, G.: Some results on the convergence of conditional distributions. Stat. Probab. Lett. 78, 3249–3253 (2008)
Acknowledgments
My sincere thanks to the referee for his valuable comments. I am also grateful the University College of Zayed University, Dubai, UAE for all supports and facilities I had.
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Rahimov, I. (2010). Approximation of a sum of martingale differences generated by a bootstrap branching process. In: González Velasco, M., Puerto, I., Martínez, R., Molina, M., Mota, M., Ramos, A. (eds) Workshop on Branching Processes and Their Applications. Lecture Notes in Statistics(), vol 197. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11156-3_9
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DOI: https://doi.org/10.1007/978-3-642-11156-3_9
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