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.
"Sequential Estimatin for Branching Processes with Immigration." Ann. Statist. 19 (4) 2232 - 2243, December, 1991. https://doi.org/10.1214/aos/1176348395