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 current population size is z≥1. As standard in process theory, an asymptotic analysis requires us condition on non-extinction up a finite generation n and let n→∞; however, because one, are able demonstrate that our do not satisfy classical consistency property (C-consistency). This leads define concept Q-consistency, prove Q-consistent asymptotically normal. To investigate circumstances C-consistent estimator preferable estimator, then provide two subcritical Galton–Watson processes. Our results rely combination linear operator coupling arguments, martingale methods.