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Estimation of the parameters of a branching process from migrating binomial observations

Part of: Parametric inference Applications Markov processes

Published online by Cambridge University Press:  01 July 2016

C. Jacob  and
J. Peccoud
C. Jacob*
Affiliation:
Institut National de la Recherche Agronomique
J. Peccoud*
Affiliation:
Université Joseph Fourier
*
∗ Postal address: INRA, Laboratoire de Biométrie, 78352 Jouy-en-Josas Cedex, France. Email address: Christine.Jacob@jouy.inra.fr
∗∗ Postal address: TIMC, IMAG, Faculté de médecine de Grenoble, Domaine de la Merci, 38706 La Tronche Cedex, France.
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Abstract

This paper considers a branching process generated by an offspring distribution F with mean m < ∞ and variance σ2 < ∞ and such that, at each generation n, there is an observed δ-migration, according to a binomial law Bpvn*Nnbef which depends on the total population size Nnbef. The δ-migration is defined as an emigration, an immigration or a null migration, depending on the value of δ, which is assumed constant throughout the different generations. The process with δ-migration is a generation-dependent Galton-Watson process, whereas the observed process is not in general a martingale. Under the assumption that the process with δ-migration is supercritical, we generalize for the observed migrating process the results relative to the Galton-Watson supercritical case that concern the asymptotic behaviour of the process and the estimation of m and σ2, as n → ∞. Moreover, an asymptotic confidence interval of the initial population size is given.

Keywords

Branching processesmigrationgeneration-dependence

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
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 4 , December 1998 , pp. 948 - 967
Copyright
Copyright © Applied Probability Trust 1998 

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