Maximum likelihood estimation for N-mixture models.

Area of Science:

• Ecology
• Statistical Modeling
• Biometrics

Background:

• N-mixture models are widely used for analyzing ecological count data, but their likelihood functions can be computationally complex.
• The standard formulation involves infinite sums, posing challenges for estimation and interpretation.

Purpose of the Study:

• To derive a closed-form expression for the likelihood of N-mixture models with a Poisson mixing distribution.
• To improve the computational tractability and algebraic properties of these models for ecological applications.

Main Methods:

• The infinite sum in the Poisson N-mixture likelihood was expressed using hypergeometric functions.
• Algebraic simplification of likelihood equations and formulation of concentrated likelihood were performed.
• The approach was validated using simulation studies and a real-world ecological dataset.

Main Results:

• A closed-form expression for the Poisson N-mixture likelihood was obtained, enabling accurate computation.
• The derived expression simplifies likelihood equations and aids in identifying problematic estimation cases.
• Similar closed-form results were found for negative binomial mixing distributions.

Conclusions:

• The closed-form likelihood significantly enhances the computational and analytical utility of N-mixture models.
• While extensions to other distributions were explored, the closed-form approach primarily benefits Poisson and negative binomial models.
• This work provides a more robust framework for analyzing ecological count data with N-mixture models.