We consider population-size-dependent branching processes (PSDBPs) which eventually become extinct with probability one. For these processes, we derive maximum likelihood estimators for the mean number of offspring born to individuals when the current population size is . As is standard in branching process theory, an asymptotic analysis of the estimators requires us to condition on non-extinction up to a finite generation n and let ; however, because the processes become extinct with probability one, we are able to demonstrate that our estimators do not satisfy the classical consistency property (C-consistency). This leads us to define the concept of Q-consistency, and we prove that our estimators are Q-consistent and asymptotically normal. To investigate the circumstances in which a C-consistent estimator is preferable to a Q-consistent estimator, we then provide two C-consistent estimators for subcritical Galton–Watson branching processes. Our results rely on a combination of linear operator theory, coupling arguments, and martingale methods.
Peter Braunsteins has conducted part of the work while supported by the Australian Research Council (ARC) Laureate Fellowship FL130100039 and the Netherlands Organisation for Scientific Research (NWO) through Gravitation-grant NETWORKS-024.002.003. Sophie Hautphenne would like to thank the Australian Research Council (ARC) for support through her Discovery Early Career Researcher Award DE150101044 and her Discovery Project DP200101281. Carmen Minuesa’s research has been supported by the Ministerio de Economía y Competitividad (grant MTM2015-70522-P), the Spanish State Research Agency (PID2019-108211GBI00/AEI/10.13039/501100011033), the Junta de Extremadura (grants IB16099 and GR18103) and the Fondo Europeo de Desarrollo Regional. This research was initiated while Carmen Minuesa was a visiting postdoctoral researcher at The University of Melbourne, and she is grateful for the hospitality and collaboration. She also acknowledges the ARC Centre of Excellence for Mathematical and Statistical Frontiers for partially supporting her research visit at this University. Finally, the authors thank the anonymous referees for their valuable comments.
All authors contributed equally to this work.
"Parameter estimation in branching processes with almost sure extinction." Bernoulli 28 (1) 33 - 63, February 2022. https://doi.org/10.3150/21-BEJ1332