On the existence of maximum likelihood estimates for presence-only data

Presence-only data can be used to determine resource selection and estimate a species' distribution. Maximum likelihood is a common parameter estimation method used for species distribution models. Maximum likelihood estimates, however, do not always exist for a commonly used species distribution model - the Poisson point process. We demonstrate the issue with conventional maximum likelihood mathematically, using a data example, and a simulation experiment and show alternative estimation methods. We found that when habitat preferences are strong or the number of presence-only locations is small, by chance, maximum likelihood coefficient estimates for the Poisson point process model may not exist. We found that several alternative estimation methods can produce reliable estimates, but results will depend on the chosen method. It is important to identify conditions for which maximum likelihood estimates are unlikely to be identifiable from presence-only data. In data sets where the maximum likelihood estimates do not exist, penalized likelihood and Bayesian methods will produce coefficient estimates, but these are sensitive to the choice of estimation procedure and prior or penalty term. When sample size is small or it is thought that habitat preferences are strong, we propose a suite of estimation procedures researchers can consider using.