Numerical evaluation of the transition probability of the simple birth-and-death process

TL;DR

This paper develops a numerically stable method for evaluating transition probabilities in the simple birth-and-death process, enabling more accurate maximum likelihood estimation from observed data.

Contribution

It introduces a hypergeometric function-based approach with recurrence relations to accurately compute transition probabilities, addressing ill-conditioning issues.

Findings

Successful numerical evaluation of transition probabilities

Effective maximum likelihood estimation on simulated data

Application to real population data

Abstract

The simple (linear) birth-and-death process is a widely used stochastic model for describing the dynamics of a population. When the process is observed discretely over time, despite the large amount of literature on the subject, little is known about formal estimator properties. Here we will show that its application to observed data is further complicated by the fact that numerical evaluation of the well-known transition probability is an ill-conditioned problem. To overcome this difficulty we will rewrite the transition probability in terms of a Gaussian hypergeometric function and subsequently obtain a three-term recurrence relation for its accurate evaluation. We will also study the properties of the hypergeometric function as a solution to the three-term recurrence relation. We will then provide formulas for the gradient and Hessian of the log-likelihood function and conclude the article by applying our methods for numerically computing maximum likelihood estimates in both simulated and real dataset.