On asymptotic normality of sequential estimators for branching processes with immigration

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Abstract

Consider a Galton-Watson process with immigration. This paper studies the limits of the sequential estimator, proposed by [Sriram, T.N., Basawa, I.V., and Huggins, R.M., (1991). Sequential estimation for branching processes with immigration. Ann. Statist. 19, 2232-2243.] and the modified sequential estimator, proposed by [Shete, S., Sriram, T.N., 1998. Fixed precision estimator of the offspring mean in branching processes. Stochastic Process. Appl. 77, 17-33.]. [Sriram, T.N., Basawa, I.V., Huggins, R.M., 1991. Sequential estimation for branching processes with immigration. Ann. Statist. 19, 2232-2243.] proved that the sequential estimators are asymptotically normal in the subcritical and critical cases. In this paper it is proved that the sequential estimators are asymptotically normal in the supercritical case and that the limiting distributions of the modified estimators, after being properly standardized, are normal as well.

Original languageEnglish (US)
Pages (from-to)2892-2902
Number of pages11
JournalJournal of Statistical Planning and Inference
Volume137
Issue number9
DOIs
StatePublished - Sep 1 2007

Bibliographical note

Funding Information:
The author would like to thank Professor Richard Green, an associate editor and a referee for their helpful comments. The research was supported in part by NSF grant DMS 0604176.

Copyright:
Copyright 2007 Elsevier B.V., All rights reserved.

Keywords

  • Asymptotic normality
  • Branching process
  • Modified sequential estimator
  • Sequential estimator
  • Stopping time

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