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On estimating the variance of the offspring distribution in a simple branching process

Published online by Cambridge University Press:  01 July 2016

C. C. Heyde
C. C. Heyde*
Affiliation:
Australian National University
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Abstract

A single realization {Zn, 0 ≦n≦N + 1} of a supercritical Galton-Watson process (so called) is considered and it is required to estimate the variance of the offspring distribution. A prospective estimator is proposed, where , and is shown to be strongly consistent on the non-extinction set. A central limit result and an iterated logarithm result are provided to give information on the rate of convergence of the estimator. It is also shown that the estimation results are robust in the sense that they continue to apply unchanged in the case where immigration occurs. Martingale limit theory is employed at each stage in obtaining the limit results.

Keywords

BRANCHING PROCESSESGALTON-WATSON PROCESSESESTIMATIONCONSISTENCYCENTRAL LIMIT THEOREMITERATED LOGARITHM LAWMARTINGALESROBUSTNESS
Type
Research Article
Information
Advances in Applied Probability , Volume 6 , Issue 3 , September 1974 , pp. 421 - 433
Copyright
Copyright © Applied Probability Trust 1974 

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References

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