title: Computational aspects of N-mixture models.
abstract: The N-mixture model is widely used to estimate the abundance of a population in the presence of unknown detection probability from only a set of counts subject to spatial and temporal replication (Royle, 2004, Biometrics 60, 105-115). We explain and exploit the equivalence of N-mixture and multivariate Poisson and negative-binomial models, which provides powerful new approaches for fitting these models. We show that particularly when detection probability and the number of sampling occasions are small, infinite estimates of abundance can arise. We propose a sample covariance as a diagnostic for this event, and demonstrate its good performance in the Poisson case. Infinite estimates may be missed in practice, due to numerical optimization procedures terminating at arbitrarily large values. It is shown that the use of a bound, K, for an infinite summation in the N-mixture likelihood can result in underestimation of abundance, so that default values of K in computer packages should be avoided. Instead we propose a simple automatic way to choose K. The methods are illustrated by analysis of data on Hermann's tortoise Testudo hermanni.
authors.affiliation: School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, Kent CT2 7NF, UK.
authors.forename: Emily B
authors.initials: EB
authors.lastname: Dennis
authors.affiliation: School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, Kent CT2 7NF, UK.
authors.forename: Byron J T
authors.initials: BJT
authors.lastname: Morgan
authors.affiliation: School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, Kent CT2 7NF, UK.
authors.forename: Martin S
authors.initials: MS
authors.lastname: Ridout
mesh_terms: D000465:Algorithms; D000818:Animals; D001699:Biometry / Q000379:methods*; D019352:Censuses; D003198:Computer Simulation; D003627:Data Interpretation, Statistical; D004784:Environmental Monitoring; D005602:France; D015233:Models, Statistical; D011157:Population Dynamics; D014426:Turtles / Q000502:physiology*
keywords: Abundance estimation; Method of moments; Multivariate Poisson; Multivariate negative binomial; Optimal design; Sampling; Temporal replication