Hostname: page-component-7bb8b95d7b-w7rtg Total loading time: 0 Render date: 2024-09-12T15:40:17.575Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic inference for partially observed branching processes

Part of: Markov processes Inference from stochastic processes

Published online by Cambridge University Press:  01 July 2016

Andrea Kvitkovičová  and
Victor M. Panaretos
Andrea Kvitkovičová*
Affiliation:
École Polytechnique Fédérale de Lausanne
Victor M. Panaretos*
Affiliation:
École Polytechnique Fédérale de Lausanne
*
∗ Postal address: Section de Mathématiques, École Polytechnique Fédérale de Lausanne, EPFL-SB Station 8 - Bâtiment MA, CH-1015 Lausanne, Switzerland.
∗ Postal address: Section de Mathématiques, École Polytechnique Fédérale de Lausanne, EPFL-SB Station 8 - Bâtiment MA, CH-1015 Lausanne, Switzerland.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider the problem of estimation in a partially observed discrete-time Galton-Watson branching process, focusing on the first two moments of the offspring distribution. Our study is motivated by modelling the counts of new cases at the onset of a stochastic epidemic, allowing for the facts that only a part of the cases is detected, and that the detection mechanism may affect the evolution of the epidemic. In this setting, the offspring mean is closely related to the spreading potential of the disease, while the second moment is connected to the variability of the mean estimators. Inference for branching processes is known for its nonstandard characteristics, as compared with classical inference. When, in addition, the true process cannot be directly observed, the problem of inference suffers significant further perturbations. We propose nonparametric estimators related to those used when the underlying process is fully observed, but suitably modified to take into account the intricate dependence structure induced by the partial observation and the interaction scheme. We show consistency, derive the limiting laws of the estimators, and construct asymptotic confidence intervals, all valid conditionally on the explosion set.

Keywords

Epidemic modelGalton-Watson branching processpartial observationconsistencyasymptotic distributionmartingalestable convergence

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 62M05: Markov processes
Secondary: 60J85: Applications of branching processes 92D30: Epidemiology
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 4 , December 2011 , pp. 1166 - 1190
Copyright
Copyright © Applied Probability Trust 2011 

References

Ball, F. and Donnelly, P. (1995). Strong approximations for epidemic models. Stoch. Process. Appl. 55, 1–21.CrossRefGoogle Scholar
Becker, N. G. and Hasofer, A. M. (1997). Estimation in epidemics with incomplete observations. J. R. Statist. Soc. B 59, 415–429.CrossRefGoogle Scholar
Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.CrossRefGoogle Scholar
Duby, C. and Rouault, A. (1982). Estimation non paramétrique de l'espérance et de la variance de la loi de reproduction d'un processus de ramification. Ann. Inst. H. Poincaré B. 18, 149–163.Google Scholar
Guttorp, P. (1991). Statistical Inference for Branching Processes. John Wiley, New York.Google Scholar
Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.Google Scholar
Kendall, D. G. (1956). Deterministic and stochastic epidemics in closed populations. In Proc. 3rd Berkeley Symp. on Mathematical Statistics and Probability Vol. IV, University of California Press, Berkeley, pp. 149–165.Google Scholar
Kvitkovičová, A. and Panaretos, V. M. (2010). Asymptotic inference for partially observed branching processes. Technical Report #01/10, Chair of Mathematical Statistics, EPFL.Google Scholar
Meester, R. and Trapman, P. (2006). Estimation in branching processes with restricted observations. Adv. Appl. Prob. 38, 1098–1115.CrossRefGoogle Scholar
Meester, R., de Koning, J., de Jong, M. C. M. and Diekmann, O. (2002). Modeling and real-time prediction of classical swine fever epidemics. Biometrics 58, 178–184.Google ScholarPubMed
Panaretos, V. M. (2007). Partially observed branching processes for stochastic epidemics. J. Math. Biol. 54, 645–668.CrossRefGoogle ScholarPubMed
Scott, D. J. (1978). A central limit theorem for martingales and an application to branching processes. Stoch. Process. Appl. 6, 241–252.CrossRefGoogle Scholar
Stephan, F. F. (1945). The expected value and variance of the reciprocal and other negative powers of a positive Bernoullian variate. Ann. Math. Statist. 16, 50–61.Google Scholar