Consistent estimation in subcritical birth-and-death processes

TL;DR

This paper studies parameter estimation in subcritical birth-and-death processes, revealing classical estimators' limitations and introducing new consistent estimators that converge to true parameters when conditioned on survival.

Contribution

The paper develops the first $C$-consistent estimators for subcritical birth-and-death processes, overcoming the inconsistency of classical maximum likelihood estimators.

Findings

Classical MLEs converge to $Q$-process parameters when conditioned on survival.

New $C$-consistent estimators converge to true parameters conditioned on survival.

Asymptotic normality of the proposed estimators is established.

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.