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On waiting times to populate an environment and a question of statistical inference

Part of: Inference from stochastic processes Markov processes

Published online by Cambridge University Press:  14 July 2016

F. Thomas Bruss  and
M. Slavtchova-Bojkova
F. Thomas Bruss*
Affiliation:
Université Libre de Bruxelles
M. Slavtchova-Bojkova*
Affiliation:
Vrije Universiteit Brussel
*
∗Postal address: Département de Mathématique et ISRO, Université Libre de Bruxelles, Campus Plaine CP 210, 1050 Bruxelles, Belgium. Email address: tbruss@ulb.ac.be
∗∗Postal address: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bontchev str., 1113 Sofia, Bulgaria.
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Abstract

The mathematical model we consider here is the classical Bienaymé–Galton–Watson branching process modified with immigration in the state zero.

We study properties of the waiting time to explosion of the supercritical modified process, i.e. that time until all beginning cycles which die out have disappeared. We then derive the expected total progeny of a cycle and show how higher moments can be computed. With a view to applications the main goal is to show that any statistical inference from observed cycle lengths or estimates of total progeny on the fertility rate of the process must be treated with care. As an example we discuss population experiments with trout.

Keywords

Bienaymé–Galton–Watson branching processgenerating functionscycle lengthtotal progenystatistical inferenceextinction bias

MSC classification

Primary: 60J85: Applications of branching processes
Secondary: 62M05: Markov processes
Type
Short Communications
Information
Journal of Applied Probability , Volume 36 , Issue 1 , March 1999 , pp. 261 - 267
Copyright
Copyright © Applied Probability Trust 1999 

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Footnotes

This work was partly supported by the FWO research network WO-001-96N and grant MM-704/97 of the National Science Foundation.

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