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

TL;DR

This paper characterizes the conditions under which the Lotka--Nagaev estimator for the offspring mean in supercritical Galton--Watson processes is qualitatively robust, providing insights into its stability across different offspring laws.

Contribution

It identifies the specific sets of offspring laws ensuring qualitative robustness of the estimator and establishes conditions for uniform robustness and consistency.

Findings

Qualitative robustness characterized by locally uniformly integrating sets

Uniform robustness achieved under global properties

Estimator is locally uniformly weakly consistent on these sets

Abstract

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.