Open Access
December, 1991 Sequential Estimatin for Branching Processes with Immigration
T. N. Sriram, I. V. Basawa, R. M. Huggins
Ann. Statist. 19(4): 2232-2243 (December, 1991). DOI: 10.1214/aos/1176348395


For the critical and subcritical Galton-Watson processes with immigration, it is shown that if the data were collected according to an appropriate stopping rule, the natural sequential estimator of the offspring mean $m$ is asymptotically normally distributed for each fixed $m \in (0, 1\rbrack.$ Furthermore, the sequential estimator is shown to be asymptotically normally distributed uniformly over a class of offspring distributions with $m \in (0, 1\rbrack$ bounded variance and satisfying a mild condition. These results are to be contrasted with the nonsequential approach where drastically different limit distributions are obtained for the two cases: (a) $m < 1$ (normal) and (b) $m = 1$ (nonnormal), thus leading to a singularity problem at $m = 1.$ The sequential approach proposed here avoids this singularity and unifies the two cases. The proof of the uniformity result is based on a uniform version of the well-known Anscombe's theorem.


Download Citation

T. N. Sriram. I. V. Basawa. R. M. Huggins. "Sequential Estimatin for Branching Processes with Immigration." Ann. Statist. 19 (4) 2232 - 2243, December, 1991.


Published: December, 1991
First available in Project Euclid: 12 April 2007

zbMATH: 0850.62636
MathSciNet: MR1135173
Digital Object Identifier: 10.1214/aos/1176348395

Primary: 60J80
Secondary: 62L10

Keywords: branching processes with immigration , fixed-width confidence intervals , sequential estimation , stopping time , uniform Anscombe's theorem , uniform asymptotic normality , uniform CLT

Rights: Copyright © 1991 Institute of Mathematical Statistics

Vol.19 • No. 4 • December, 1991
Back to Top