The one-inflated positive Poisson mixture model for use in population size estimation.

The one-inflated positive Poisson mixture model (OIPPMM) is presented, for use as the truncated count model in Horvitz-Thompson estimation of an unknown population size. The OIPPMM offers a way to address two important features of some capture-recapture data: one-inflation and unobserved heterogeneity. The OIPPMM provides markedly different results than some other popular estimators, and these other estimators can appear to be quite biased, or utterly fail due to the boundary problem, when the OIPPMM is the true data-generating process. In addition, the OIPPMM provides a solution to the boundary problem, by labelling any mixture components on the boundary instead as one-inflation.

One-inflation and unobserved heterogeneity in population size estimation.

We present the one-inflated zero-truncated negative binomial (OIZTNB) model, and propose its use as the truncated count distribution in Horvitz-Thompson estimation of an unknown population size. In the presence of unobserved heterogeneity, the zero-truncated negative binomial (ZTNB) model is a natural choice over the positive Poisson (PP) model; however, when one-inflation is present the ZTNB model either suffers from a boundary problem, or provides extremely biased population size estimates. Monte Carlo evidence suggests that in the presence of one-inflation, the Horvitz-Thompson estimator under the ZTNB model can converge in probability to infinity. The OIZTNB model gives markedly different population size estimates compared to some existing truncated count distributions, when applied to several capture-recapture data that exhibit both one-inflation and unobserved heterogeneity.