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On estimation of the variances for critical branching processes with immigration

Part of: Special processes Stochastic analysis Markov processes

Published online by Cambridge University Press:  14 July 2016

Chunhua Ma  and
Longmin Wang
Chunhua Ma*
Affiliation:
Nankai University
Longmin Wang*
Affiliation:
Nankai University
*
∗Postal address: School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, P. R. China.
∗Postal address: School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, P. R. China.
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Abstract

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The conditional least-squares estimators of the variances are studied for a critical branching process with immigration that allows the offspring distributions to have infinite fourth moments. We derive different forms of limiting distributions for these estimators when the offspring distributions have regularly varying tails with index α. In particular, in the case in which 2 < α < 8/3, the normalizing factor of the estimator for the offspring variance is smaller than √n, which is different from that of Winnicki (1991).

Keywords

Branching process with immigrationregular variationconditional least squares estimatorlimiting distribution

MSC classification

Secondary: 60J35: Transition functions, generators and resolvents 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60H20: Stochastic integral equations 60K37: Processes in random environments
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 2 , June 2010 , pp. 526 - 542
Copyright
Copyright © Applied Probability Trust 2010 

Footnotes

Supported by the NSFC (grant numbeers 10871103 and 10971106)

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