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Bayesian analysis for controlled branching processes. (English) Zbl 1354.60092
del Puerto, Inés M. (ed.) et al., Branching processes and their applications. Proceedings of the workshop, WBPA, Badajoz, Spain, April 7–10, 2015. Cham: Springer (ISBN 978-3-319-31639-0/pbk; 978-3-319-31641-3/ebook). Lecture Notes in Statistics 219, 185-205 (2016).
Summary: The branching model considered in the present work is the controlled branching process. This model is a generalization of the standard Bienaymé-Galton-Watson (BGW) branching process, and, in the terminology of population dynamics, is used to describe the evolution of populations in which a control of the population size at each generation is needed. This control consists of determining mathematically the number of individuals with reproductive capacity at each generation through a random process. In practice, this branching model can describe reasonably well the probabilistic evolution of populations in which, for various reasons of an environmental, social, or other nature, there is a mechanism that establishes the number of progenitors which take part in each generation. For example, in an ecological context, one can think of an invasive animal species that is widely recognized as a threat to native ecosystems, but there is disagreement about plans to eradicate it, i.e., while the presence of the species is appreciated by a part of the society, if its numbers are left uncontrolled it is known to be very harmful to native ecosystems. In such a case, it is better to control the population to keep it within admissible limits even though this might mean periods when animals have to be culled. Another practical situation that can be modelled by this kind of process is the evolution of an animal population that is threatened by the existence of predators. In each generation, the survival of each animal (and therefore the possibility of giving new births) will be strongly affected by this factor, making the introduction of a random mechanism necessary to model the evolution of this kind of population.
For the entire collection see [Zbl 1354.60003].

MSC:
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
62C10 Bayesian problems; characterization of Bayes procedures
60J85 Applications of branching processes
92D25 Population dynamics (general)
Software:
CODA; R; Rmpi; snow; Snow
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References:
[1] Bagley, J.H.: On the almost sure convergence of controlled branching processes. J. Appl. Probab. 23, 827–831 (1986) · Zbl 0608.60079 · doi:10.1017/S0021900200111970
[2] Brooks, S.: Markov Chain Monte Carlo method and its application. J. R. Stat. Soc. Ser. D (The Statistician) 47, 69–100 (1998) · doi:10.1111/1467-9884.00117
[3] Brooks, S., Gelman, A.: General methods for monitoring convergence of iterative simulations. J. Comput. Graph. Stat. 7, 434–455 (1998)
[4] Gelman, A., Rubin, D.: Inference from iterative simulation using multiple sequences. Stat. Sci. 7, 457–511 (1992) · Zbl 1386.65060 · doi:10.1214/ss/1177011136
[5] González, M., Molina, M., del Puerto, I.: On the class of controlled branching process with random control functions. J. Appl. Probab. 39, 804–815 (2002) · Zbl 1032.60077 · doi:10.1017/S0021900200022051
[6] González, M., Martínez, R., del Puerto, I.: Nonparametric estimation of the offspring distribution and mean for a controlled branching process. Test 13, 465–479 (2004) · Zbl 1069.62064 · doi:10.1007/BF02595782
[7] González, M., Martínez, R., del Puerto, I.: Estimation of the variance for a controlled branching process. Test 14, 199–213 (2005) · Zbl 1069.62063 · doi:10.1007/BF02595403
[8] González, M., Molina, M., del Puerto, I.: Asymptotic behaviour of critical controlled branching process with random control function. J. Appl. Probab. 42 (2), 463–477 (2005) · Zbl 1079.60073 · doi:10.1017/S0021900200000462
[9] González, M., Martín, J., Martínez, R., Mota, M.: Non-parametric Bayesian estimation for multitype branching processes through simulation-based methods. Comput. Stat. Data Anal. 52, 1281–1291 (2008) · Zbl 1332.62116 · doi:10.1016/j.csda.2007.06.008
[10] González, M., Gutiérrez, C., Martínez, R., del Puerto, I.: Bayesian inference for controlled branching processes through MCMC and ABC methodologies. Rev. R. Acad. Cien. Serie A. Mat. 107, 459–473 (2013) · Zbl 1277.62083 · doi:10.1007/s13398-012-0072-8
[11] González, M., Minuesa, C., del Puerto, I.: Maximum likelihood estimation and Expectation-Maximization algorithm for controlled branching process. Comput. Stat. Data Anal. 93, 209–227 (2016) · Zbl 06918698 · doi:10.1016/j.csda.2015.01.015
[12] Jagers, P.: Branching Processes with Biological Applications. Wiley, London (1975) · Zbl 0356.60039
[13] Martínez, R., Mota, M., del Puerto, I.: On asymptotic posterior normality for controlled branching processes with random control function. Statistics 43, 367–378 (2009) · Zbl 1278.60133 · doi:10.1080/02331880802395971
[14] Mendoza, M., Gutiérrez-Peña, E.: Bayesian conjugate analysis of the Galton-Watson process. Test 9, 149–171 (2000) · Zbl 0954.62027 · doi:10.1007/BF02595856
[15] Plummer, M., Best, N., Cowles, K., Vines, K.: coda: Output analysis and diagnostics for MCMC. R package version 0.13-5 (2010)
[16] R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2014)
[17] Rahimov, I.: Limit distributions for weighted estimators of the offspring mean in a branching process. Test 18, 568–583 (2009) · Zbl 1203.60131 · doi:10.1007/s11749-008-0124-8
[18] Sevastyanov, B.A., Zubkov, A.: Controlled branching processes. Theor. Prob. Appl. 19, 14–24 (1974) · Zbl 0308.60051 · doi:10.1137/1119002
[19] Sriram, T., Bhattacharya, A., González, M., Martínez, R., del Puerto, I.: Estimation of the offspring mean in a controlled branching process with a random control function. Stoch. Process. Appl. 117, 928–946 (2007) · Zbl 1114.62082 · doi:10.1016/j.spa.2006.11.002
[20] Tierney, L.: Markov chains for exploring posterior distributions. Ann. Stat. 22, 1701–1762 (1994) · Zbl 0829.62080 · doi:10.1214/aos/1176325750
[21] Tierney, L., Rossini, A.J., Li, N., Sevcikova, H.: snow: Simple Network of Workstations. R package version 0.3-3 (2013)
[22] Tjøstheim, D.: Some recent theory for autoregressive count time series. Test 21, 413–438 (2012) · Zbl 1362.62174 · doi:10.1007/s11749-012-0296-0
[23] Weiß, C.H.: Thinning operations for modelling time series of counts- a survey. Adv. Stat. Anal. 92, 319–341 (2008) · doi:10.1007/s10182-008-0072-3
[24] Weiß, C.H.: Fully observed INAR(1) processes. J. Appl. Stat. 39, 581–598 (2011) · doi:10.1080/02664763.2011.604308
[25] Yu, H.: Rmpi: parallel statistical computing in R. R News 2, 10–14 (2002)
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