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Parameter estimation in branching processes with almost sure extinction

We consider population-size-dependent branching processes (PSDBPs) which eventually become extinct with probability one. For these processes, we derive maximum likelihood estimators for the mean number of offspring born to individuals when the current population size is $z\geq 1$. As is standard in branching process theory, an asymptotic analysis of the estimators requires us to condition on non-extinction up to a finite generation $n$ and let $n\to\infty$; however, because the processes become extinct with probability one, we are able to demonstrate that our estimators do not satisfy the classical consistency property ($C$-consistency). This leads us to define the concept of $Q$-consistency, and we prove that our estimators are $Q$-consistent and asymptotically normal. To investigate the circumstances in which a $C$-consistent estimator is preferable to a $Q$-consistent estimator, we then provide two $C$-consistent estimators for subcritical Galton-Watson branching processes. Our results rely on a combination of linear operator theory, coupling arguments, and martingale methods.