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 language | English (US) |
|---|---|
| Pages (from-to) | 2892-2902 |
| Number of pages | 11 |
| Journal | Journal of Statistical Planning and Inference |
| Volume | 137 |
| Issue number | 9 |
| DOIs | |
| State | Published - 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.
Keywords
- Asymptotic normality
- Branching process
- Modified sequential estimator
- Sequential estimator
- Stopping time