On qualitative robustness of the Lotka--Nagaev estimator for the offspring mean of a supercritical Galton--Watson process

We characterize the sets of offspring laws on which the Lotka--Nagaev estimator for the mean of a supercritical Galton--Watson process is qualitatively robust. These are exactly the locally uniformly integrating sets of offspring laws, which may be quite large. If the corresponding global property is assumed instead, we obtain uniform robustness as well. We illustrate both results with a number of concrete examples. As a by-product of the proof we obtain that the Lotka--Nagaev estimator is [locally] uniformly weakly consistent on the respective sets of offspring laws, conditionally on non-extinction.