Consistent least squares estimation in population-size-dependent branching processes

TL;DR

This paper introduces new conditionally consistent estimators for population-size-dependent branching processes, addressing extinction bias and applying to endangered species, with proven asymptotic properties and practical case studies.

Contribution

It develops the first conditionally consistent estimators for logistic growth population models based on single trajectories, with proofs of consistency and asymptotic normality.

Findings

Estimators reduce observation bias in endangered populations.

Simulated examples demonstrate estimator effectiveness.

Application to Chatham Island black robin estimates carrying capacity.

Abstract

We derive the first conditionally consistent estimators for a class of parametric Markov population models with logistic growth, which are suitable for modelling endangered populations in restricted habitats with a carrying capacity. We focus on discrete-time parametric population-size-dependent branching processes, for which we propose a new class of weighted least-squares estimators based on a single trajectory of population size counts. We establish the consistency and asymptotic normality of our estimators, conditional on non-extinction up to time $n$, as $n\to\infty$. Since Markov population models with a carrying capacity become extinct almost surely under general conditions, our proofs rely on arguments distinct from those in the existing literature. Our results are motivated by conservation biology, where endangered populations are often studied precisely because they are still alive, leading to an observation bias. Through simulated examples, we show that our conditionally consistent estimators generally reduce this bias for key quantities such as a habitat's carrying capacity. We apply our methodology to estimate the carrying capacity of the Chatham Island black robin, a population reduced to a single breeding female in the 1970's, which has since recovered but has yet to reach the island's carrying capacity.