Estimating population size: The importance of model and estimator choice

We consider estimator and model choice when estimating abundance from capture-recapture data. Our work is motivated by a mark-recapture distance sampling example, where model and estimator choice led to unexpectedly large disparities in the estimates. To understand these differences, we look at three estimation strategies (maximum likelihood estimation, conditional maximum likelihood estimation, and Bayesian estimation) for both binomial and Poisson models. We show that assuming the data have a binomial or multinomial distribution introduces implicit and unnoticed assumptions that are not addressed when fitting with maximum likelihood estimation. This can have an important effect in finite samples, particularly if our data arise from multiple populations. We relate these results to those of restricted maximum likelihood in linear mixed effects models.