Consistent estimation in subcritical birth-and-death processes

Sophie Hautphenne; Emma Horton

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arXiv · 2511.01153

Consistent estimation in subcritical birth-and-death processes

Sophie Hautphenne·Emma Horton

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Abstract

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 $t \to \infty$ 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.

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Sophie Hautphenne, Emma Horton. 2025-11-03. Consistent estimation in subcritical birth-and-death processes. https://arxiv.org/abs/2511.01153

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