Asymptotic Inference for Partially Observed Branching Processes

KVITKOVIČOVÁ, Andrea and VM PANARETOS. Asymptotic Inference for Partially Observed Branching Processes. Advances in Applied Probability. Sheffield: Applied Probability Trust, 2011, vol. 43, No 4, p. 1166-1190. ISSN 0001-8678.

Basic information

• Original name: Asymptotic Inference for Partially Observed Branching Processes
• Authors: KVITKOVIČOVÁ, Andrea and VM PANARETOS.
• Edition: Advances in Applied Probability, Sheffield, Applied Probability Trust, 2011, 0001-8678.

Other information

• Original language: English
• Type of outcome: Article in a journal
• Confidentiality degree: is not subject to a state or trade secret
• Impact factor: Impact factor: 0.679
• UT WoS: 000298713900012
• Keywords in English: Epidemic model; Galton-Watson branching process; partial observation; consistency; asymptotic distribution; martingale; stable convergence
• Tags: International impact, Reviewed

Abstract

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.