Consistent estimation in subcritical birth-and-death processes

We investigate parameter estimation in subcritical continuous-time birth-and-death processes with multiple births. We show that the classical maximum likelihood estimators for the model parameters, based on the continuous observation of a single non-extinct trajectory, are not consistent in the usual sense: conditional on survival up to time t , they converge as t → ∞ to the corresponding quantities in the associated Q -process, namely the process conditioned to survive in the distant future. We develop the first C-consistent estimators in this setting, which converge to the true parameter values when conditioning on survival up to time t , and establish their asymptotic normality. The analysis relies on spine decompositions and coupling techniques.