  Estimating the size of an elusive target population is of prominent interest
in many areas in the life and social sciences. Our aim is to provide an
efficient and workable method to estimate the unknown population size, given
the frequency distribution of counts of repeated identifications of units of
the population of interest. This counting variable is necessarily
zero-truncated, since units that have never been identified are not in the
sample. We consider several applications: clinical medicine, where interest is
in estimating patients with adenomatous polyps which have been overlooked by
the diagnostic procedure; drug user studies, where interest is in estimating
the number of hidden drug users which are not identified; veterinary
surveillance of scrapie in the UK, where interest is in estimating the hidden
amount of scrapie; and entomology and microbial ecology, where interest is in
estimating the number of unobserved species of organisms. In all these
examples, simple models such as the homogenous Poisson are not appropriate
since they do not account for present and latent heterogeneity. The
Poisson-Gamma (negative binomial) model provides a flexible alternative and
often leads to well-fitting models. It has a long history and was recently used
in the development of the Chao-Bunge estimator. Here we use a different
property of the Poisson-Gamma model: if we consider ratios of neighboring
Poisson-Gamma probabilities, then these are linearly related to the counts of
repeated identifications.
