Advances in Applied Probability

Asymptotic inference for partially observed branching processes

Andrea Kvitkovičová and Victor M. Panaretos

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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.

Article information

Adv. in Appl. Probab., Volume 43, Number 4 (2011), 1166-1190.

First available in Project Euclid: 16 December 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 62M05: Markov processes: estimation
Secondary: 60J85: Applications of branching processes [See also 92Dxx] 92D30: Epidemiology

Epidemic model Galton-Watson branching process partial observation consistency asymptotic distribution martingale stable convergence


Kvitkovičová, Andrea; Panaretos, Victor M. Asymptotic inference for partially observed branching processes. Adv. in Appl. Probab. 43 (2011), no. 4, 1166--1190. doi:10.1239/aap/1324045703.

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