An Approximate Maximum Likelihood Estimator for Discretely Observed Linear Birth-and-Death Processes

Linear birth-and-death processes (LBDPs) are foundational stochastic models in population dynamics, evolutionary biology, and hematopoiesis. Estimating parameters from discretely observed data is computationally demanding due to irregular sampling, noise, and missing values. We propose a novel approximate maximum likelihood estimator (MLE) for LBDPs based on a Gaussian approximation to transition probabilities. The approach transforms estimation into a univariate optimization problem, achieving substantial computational gains without sacrificing accuracy. Through simulations, we show that the approximate MLE outperforms Gaussian and saddlepoint-based estimators in speed and precision under realistic noise and sparsity. Applied to longitudinal clonal hematopoiesis data, the method produces biologically meaningful growth estimates even with noisy, compositional input. Unlike Gaussian and saddlepoint approximations, our estimator is invariant to data scaling, making it ideal for real-world applications such as variant allele frequency analyses.