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.