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.
Publié le : 1991-12-14
Classification:
Fixed-width confidence intervals,
sequential estimation,
branching processes with immigration,
stopping time,
uniform CLT,
uniform Anscombe's theorem,
uniform asymptotic normality,
60J80,
62L10
@article{1176348395,
author = {Sriram, T. N. and Basawa, I. V. and Huggins, R. M.},
title = {Sequential Estimatin for Branching Processes with Immigration},
journal = {Ann. Statist.},
volume = {19},
number = {1},
year = {1991},
pages = { 2232-2243},
language = {en},
url = {http://dml.mathdoc.fr/item/1176348395}
}
Sriram, T. N.; Basawa, I. V.; Huggins, R. M. Sequential Estimatin for Branching Processes with Immigration. Ann. Statist., Tome 19 (1991) no. 1, pp. 2232-2243. http://gdmltest.u-ga.fr/item/1176348395/