Advances in Applied Probability

Asymptotic inference for partially observed branching processes

Andrea Kvitkovičová and Victor M. Panaretos

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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.

Article information

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

Dates
First available in Project Euclid: 16 December 2011

Permanent link to this document
https://projecteuclid.org/euclid.aap/1324045703

Digital Object Identifier
doi:10.1239/aap/1324045703

Mathematical Reviews number (MathSciNet)
MR2867950

Zentralblatt MATH identifier
1230.62108

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.aap/1324045703


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