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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.
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Abstract

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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 

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