Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-23T11:10:38.292Z Has data issue: false hasContentIssue false
  • English
  • Français

Statistical inference for partially observed branching processes with immigration

Part of: Markov processes Stochastic processes Parametric inference

Published online by Cambridge University Press:  04 April 2017

Ibrahim Rahimov
Ibrahim Rahimov*
Affiliation:
Zayed University
*
* Postal address: Department of Mathematics and Statistics, Zayed University, Box 19282, Dubai, UAE. Email address: ibrahim.rahimov@zu.ac.ae
Get access Rights & Permissions [Opens in a new window]

Abstract

In the paper we consider the following modification of a discrete-time branching process with stationary immigration. In each generation a binomially distributed subset of the population will be observed. The number of observed individuals constitute a partially observed branching process. After inspection both observed and unobserved individuals may change their offspring distributions. In the subcritical case we investigate the possibility of using the known estimators for the offspring mean and for the mean of the stationary-limiting distribution of the process when the observation of the population sizes is restricted. We prove that, if both the population and the number of immigrants are partially observed, the estimators are still strongly consistent. We also prove that the `skipped' version of the estimator for the offspring mean is asymptotically normal and the estimator of the stationary distribution's mean is asymptotically normal under additional assumptions.

Keywords

Branching processsubcriticalimmigrationrestricted observationoffspring meanstationary distributionrandom sumlimit theorem

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 62F12: Asymptotic properties of estimators 60G99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 1 , March 2017 , pp. 82 - 95
Copyright
Copyright © Applied Probability Trust 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andersson, H. and Britton, T. (2000).Stochastic Epidemic Models and their Statistical Analysis.Springer,New York.Google Scholar
Billingsley, P. (1999).Convergence of Probability Measures, 2nd edn.John Wiley,New York.Google Scholar
Gut, A. (2005).Probability: A Graduate Course, 2nd edn.Springer,New York.Google Scholar
Hall, P. and Heyde, C. C. (1980).Martingale Limit Theory and Its Application.Academic Press,New York.Google Scholar
Knopp, K. (1956).Infinite Sequences and Series.Dover,New York.Google Scholar
Kvitkovičová, A. and Panaretos, V. M. (2011).Asymptotic inference for partially observed branching processes.Adv. Appl. Prob. 43,1166–1190.Google Scholar
Meester, R. and Trapman, P. (2006).Estimation in branching processes with restricted observations.Adv. Appl. Prob. 38,1098–1115.Google Scholar
Meester, R., De Koning, J., De Jong, M. S. and Diekmann, O. (2002).Modeling and real-time prediction of classical swine fever epidemics.Biometrics 58,178–184.Google Scholar
Nanthi, K. (1979).Some limit theorems of statistical relevance on branching processes. Doctoral Thesis, University of Madras.Google Scholar
Pakes, A. G. (1971).Branching processes with immigration.J. Appl. Prob. 8,32–42.Google Scholar
Panaretos, V. M. (2007).Partially observed branching processes for stochastic epidemics.J. Math. Biol. 54,645–668.Google Scholar
Petrov, V. V. (1995).Limit Theorems of Probability Theory (Oxford Studies in Probability 4).Oxford University Press.Google Scholar
Scott, D. J. (1977/78).A central limit theorem for martingales and an application to branching processes.Stoch. Process. Appl. 6,241–252.Google Scholar
Sriram, T. N. (1991).On uniform strong consistency of an estimator of the offspring mean in a branching process with immigration.Statist. Prob. Lett. 12,151–155.Google Scholar
Venkataraman, K. N. and Nanthi, K. (1982).A limit theorem on subcritical Galton–Watson process with immigration.Ann. Prob. 10,1069–1074.Google Scholar
Yanev, N. M. (2008).Statistical inference for branching processes. In Records and Branching Processes, eds M. Ahsanullah and G. P. Yanev, Nova Science,New York.Google Scholar