Consistent estimation in subcritical birth-and-death processes

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.

Figures (6)

• Figure 1: Binary case. Asymptotic bias of the classical MLEs $\hat{\lambda}_t$ (left) and $\hat{\mu}_t$ (right), as functions of pairs $(\lambda,\mu)$ with $\lambda<\mu$ (subcritical case).
• Figure 2: Multiple-birth case ($Z_0=5$). Median estimates of the parameters $\lambda$, $\mu$, and $m$ as functions of the observation time $t$, based on 1500 simulated trajectories with initial population size $Z_0=5$. Each curve corresponds to a different estimator: $C$-consistent ($\tilde{\lambda}_t,\tilde{\mu}_t$), $Q$-consistent ($\hat{\lambda}_t,\hat{\mu}_t$), skeleton $h$ (the $\delta$-skeleton approach with step size $h$), and MLE $Q$-process (version 1: $m$ estimated via fixed point; version 2: finite offspring support, $m$ estimated via the $p_k$'s). Grey dashed lines indicate the true parameter values.
• Figure 3: Multiple-birth case ($Z_0=5$). Median estimates of the offspring probabilities $p_2,p_3,p_4$ as functions of the observation time $t$, based on 1500 simulated trajectories with $Z_0=5$. Three variants are compared: $C$-consistent $\tilde{p}_{k,t}$, $Q$-consistent $\hat{p}_{k,t}$, and the $C$-consistent normalised 'multinomial' version $\bar{p}_{k,t}$. Grey dashed lines indicate the true parameter values.
• Figure 4: Multiple-birth case ($Z_0=5$). Mean squared errors (MSE) of the estimators of (a) $\lambda$ and (b) $p_2$ as functions of $t$.
• Figure 5: Multiple-birth case ($Z_0=200$). Mean squared errors (MSE) of the estimators of (a) $\lambda$ and (b) $p_2$ as functions of $t$.

Theorems & Definitions (16)

• Lemma 1
• Proposition 1: $Q$-consistency of $\hat{\lambda}_t$ and $\hat{\mu}_t$
• Remark 1
• Proposition 2: Consistency of $\hat{\lambda}^\uparrow_t$ and $\hat{\mu}^\uparrow_t$
• Theorem 1: $C$-consistency of $\tilde{\lambda}_t$ and $\tilde{\mu}_t$
• Theorem 2: Asymptotic normality of $\tilde{\lambda}_t$ and $\tilde{\mu}_t$
• Corollary 1
• Theorem 3: $C$-consistency and asymptotic normality of $\tilde{p}_{k,t}$
• Corollary 2: $Q$-consistency of $\hat{p}_{k,t}$
• Remark 2