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Estimation in branching processes with restricted observations

Part of: Parametric inference Markov processes Applications Stochastic processes

Published online by Cambridge University Press:  08 September 2016

Ronald Meester  and
Pieter Trapman
Ronald Meester*
Affiliation:
Vrije Universiteit Amsterdam
Pieter Trapman*
Affiliation:
Utrecht University and Vrije Universiteit Amsterdam
*
∗ Postal address: Department of Mathematics, Vrije Universiteit Amsterdam, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands. Email address: [email protected]
∗∗ Current address: Julius Center, University Medical Center Utrecht, PO Box 85500, 3508 GA Utrecht, The Netherlands. Email address: [email protected]
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Abstract

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We consider an epidemic model where the spread of the epidemic can be described by a discrete-time Galton-Watson branching process. Between times n and n + 1, any infected individual is detected with unknown probability π and the numbers of these detected individuals are the only observations we have. Detected individuals produce a reduced number of offspring in the time interval of detection, and no offspring at all thereafter. If only the generation sizes of a Galton-Watson process are observed, it is known that one can only estimate the first two moments of the offspring distribution consistently on the explosion set of the process (and, apart from some lattice parameters, no parameters that are not determined by those moments). Somewhat surprisingly, in our context, where we observe a binomially distributed subset of each generation, we are able to estimate three functions of the parameters consistently. In concrete situations, this often enables us to estimate π consistently, as well as the mean number of offspring. We apply the estimators to data for a real epidemic of classical swine fever.

Keywords

Galton-Watson branching processconsistent estimationpartial observationepidemicreal-time estimation

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 62F12: Asymptotic properties of estimators
Secondary: 62P10: Applications to biology and medical sciences 60G42: Martingales with discrete parameter
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 4 , December 2006 , pp. 1098 - 1115
Copyright
Copyright © Applied Probability Trust 2006 

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