On the integrated maximum likelihood estimators for a closed population capture–recapture model with unequal capture probabilities

Abstract

Nuisance parameter elimination is a central problem in capture–recapture modelling. In this paper, we consider a closed population capture–recapture model which assumes the capture probabilities varies only with the sampling occasions. In this model, the capture probabilities are regarded as nuisance parameters and the unknown number of individuals is the parameter of interest. In order to eliminate the nuisance parameters, the likelihood function is integrated with respect to a weight function (uniform and Jeffrey's) of the nuisance parameters resulting in an integrated likelihood function depending only on the population size. For these integrated likelihood functions, analytical expressions for the maximum likelihood estimates are obtained and it is proved that they are always finite and unique. Variance estimates of the proposed estimators are obtained via a parametric bootstrap resampling procedure. The proposed methods are illustrated on a real data set and their frequentist properties are assessed by means of a simulation study.

Keywords:

• capture–recapture
• integrated likelihood function
• maximum likelihood estimate
• population size

Acknowledgements

The authors thank to the anonymous Referees for their comments and suggestions. The research is partially supported by the Brazilian organizations CNPq and FAPESP.