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Maximum likelihood estimation and expectation–maximization algorithm for controlled branching processes

Author

Listed:
  • González, M.
  • Minuesa, C.
  • del Puerto, I.

Abstract

The controlled branching process is a generalization of the classical Bienaymé–Galton–Watson branching process. It is a useful model for describing the evolution of populations in which the population size at each generation needs to be controlled. The maximum likelihood estimation of the parameters of interest for this process is addressed under various sample schemes. Firstly, assuming that the entire family tree can be observed, the corresponding estimators are obtained and their asymptotic properties investigated. Secondly, since in practice it is not usual to observe such a sample, the maximum likelihood estimation is initially considered using the sample given by the total number of individuals and progenitors of each generation, and then using the sample given by only the generation sizes. Expectation–maximization algorithms are developed to address these problems as incomplete data estimation problems. The accuracy of the procedures is illustrated by means of a simulated example.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:csdana:v:93:y:2016:i:c:p:209-227
    DOI: 10.1016/j.csda.2015.01.015
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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. 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.
    3. 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.
    4. Hautphenne, Sophie & Fackrell, Mark, 2014. "An EM algorithm for the model fitting of Markovian binary trees," Computational Statistics & Data Analysis, Elsevier, vol. 70(C), pages 19-34.
    5. Wang, Naichen & Wang, Lianming & McMahan, Christopher S., 2015. "Regression analysis of bivariate current status data under the Gamma-frailty proportional hazards model using the EM algorithm," Computational Statistics & Data Analysis, Elsevier, vol. 83(C), pages 140-150.
    6. Bercu, B., 1999. "Weighted estimation and tracking for Bienaymé Galton Watson processes with adaptive control," Statistics & Probability Letters, Elsevier, vol. 42(4), pages 415-421, May.
    7. Bernhardt, Paul W. & Zhang, Daowen & Wang, Huixia Judy, 2015. "A fast EM algorithm for fitting joint models of a binary response and multiple longitudinal covariates subject to detection limits," Computational Statistics & Data Analysis, Elsevier, vol. 85(C), pages 37-53.
    8. Veen, Alejandro & Schoenberg, Frederic P., 2008. "Estimation of SpaceTime Branching Process Models in Seismology Using an EMType Algorithm," Journal of the American Statistical Association, American Statistical Association, vol. 103, pages 614-624, June.
    9. 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. Nina Daskalova, 2017. "Expectation maximization estimates of the offspring probabilities in a class of multitype branching processes with binary family trees," Mathematical Population Studies, Taylor & Francis Journals, vol. 24(4), pages 246-256, October.
    2. Ramtirthkar, Mukund & Kale, Mohan, 2022. "A note on the local asymptotic mixed normality of a controlled branching process with a random control function," Statistics & Probability Letters, Elsevier, vol. 181(C).

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