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Estimation of the offspring mean in a controlled branching process with a random control function

Author

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  • Sriram, T.N.
  • Bhattacharya, A.
  • González, M.
  • Martínez, R.
  • del Puerto, I.

Abstract

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 the work of Wei and Winnicki [C.Z. Wei, J. Winnicki, Estimation of the mean in the branching process with immigration, Ann. Statist. 18 (1990) 1757-1773], 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 [C.Z. Wei, J. Winnicki, Estimation of the mean in the branching process with immigration, Ann. Statist. 18 (1990) 1757-1773] for branching processes with immigration and provide a unified limit theory of estimation.

Suggested Citation

  • Sriram, T.N. & Bhattacharya, A. & González, M. & Martínez, R. & del Puerto, I., 2007. "Estimation of the offspring mean in a controlled branching process with a random control function," Stochastic Processes and their Applications, Elsevier, vol. 117(7), pages 928-946, July.
    • Handle: RePEc:eee:spapps:v:117:y:2007:i:7:p:928-946
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    References listed on IDEAS

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    1. Miguel González & Rodrigo Martínez & Inés Puerto, 2004. "Nonparametric estimation of the offspring distribution and the mean for a controlled branching process," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 13(2), pages 465-479, December.
    2. González, M. & Molina, M. & del Puerto, I., 2004. "Limiting distribution for subcritical controlled branching processes with random control function," Statistics & Probability Letters, Elsevier, vol. 67(3), pages 277-284, April.
    3. Qi, Yongcheng & Reeves, Jaxk, 0. "On sequential estimation for branching processes with immigration," Stochastic Processes and their Applications, Elsevier, vol. 100(1-2), pages 41-51, July.
    4. Wei, C. Z. & Winnicki, J., 1989. "Some asymptotic results for the branching process with immigration," Stochastic Processes and their Applications, Elsevier, vol. 31(2), pages 261-282, April.
    5. Miguel González & Rodrigo Martínez & Iné Puerto, 2005. "Estimation of the variance for a controlled branching process," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 14(1), pages 199-213, June.
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    Cited by:

    1. M. González & M. Mota & I. Puerto, 2011. "Weighted conditional least square estimators for bisexual branching processes with immigration," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(3), pages 607-629, November.
    2. González, M. & Minuesa, C. & del Puerto, I., 2016. "Maximum likelihood estimation and expectation–maximization algorithm for controlled branching processes," Computational Statistics & Data Analysis, Elsevier, vol. 93(C), pages 209-227.
    3. Miguel González & Inés M. Puerto, 2012. "Diffusion Approximation of an Array of Controlled Branching Processes," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 843-861, September.

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