Parameter estimation for discretely-observed linear birth-and-death processes

TL;DR

This paper develops a novel method for estimating parameters of linear birth-and-death processes from periodic data, addressing numerical stability issues and providing theoretical guarantees.

Contribution

It introduces a saddlepoint approximation-based maximum likelihood estimation approach for discretely observed birth-death processes, with proven consistency and asymptotic normality.

Findings

Estimator performs well on numerical examples

Method applied successfully to real census and pandemic data

Gaussian approximation links different estimation approaches

Abstract

Birth-and-death processes are widely used to model the development of biological populations. Although they are relatively simple models, their parameters can be challenging to estimate, because the likelihood can become numerically unstable when data arise from the most common sampling schemes, such as annual population censuses. Simple estimators may be based on an embedded Galton-Watson process, but this presupposes that the observation times are equi-spaced. We estimate the birth, death, and growth rates of a linear birth-and-death process whose population size is periodically observed via an embedded Galton-Watson process, and by maximizing a saddlepoint approximation to the likelihood. We show that a Gaussian approximation to the saddlepoint-based likelihood connects the two approaches, we establish consistency and asymptotic normality of quasi-likelihood estimators, compare our estimators on some numerical examples, and apply our results to census data for two endangered bird populations and the H1N1 influenza pandemic.