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Diffusion Approximation of an Array of Controlled Branching Processes

Author

Listed:
  • Miguel González

    (University of Extremadura)

  • Inés M. Puerto

    (University of Extremadura)

Abstract

In this paper an array of controlled branching processes is considered. Using operator semigroup convergence theorems, it is proved that the fluctuation limit is a diffusion process under the conditions that the offspring and control means tend to be critical. As an application of this result, in a parametric framework, it is obtained that the bootstrapping distribution of the weighted conditional least squares estimator of the offspring mean in the critical case is not consistent. From this, it is concluded that the standard parametric bootstrap weighted conditional least squares estimate is asymptotically invalid in the critical case.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:14:y:2012:i:3:d:10.1007_s11009-012-9285-8
    DOI: 10.1007/s11009-012-9285-8
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    References listed on IDEAS

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    1. 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.
    2. Rahimov, I., 2009. "Bootstrap of the offspring mean in the critical process with a non-stationary immigration," Stochastic Processes and their Applications, Elsevier, vol. 119(11), pages 3939-3954, November.
    3. 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.
    4. Datta, Somnath & Sriram, T. N., 1995. "A modified bootstrap for branching processes with immigration," Stochastic Processes and their Applications, Elsevier, vol. 56(2), pages 275-294, April.
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    Cited by:

    1. 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.
    2. Liu, Jiawei, 2024. "A scaling limit of controlled branching processes," Statistics & Probability Letters, Elsevier, vol. 208(C).

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    5. 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.
    6. Rahimov, I., 2009. "Bootstrap of the offspring mean in the critical process with a non-stationary immigration," Stochastic Processes and their Applications, Elsevier, vol. 119(11), pages 3939-3954, November.

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