Estimating the observable population size from biased samples: a new   approach to population estimation with capture heterogeneity
  Capture-recapture methods aim to estimate the size of a closed population on the basis of multiple incomplete enumerations of individuals. In many applications, the individual probability of being recorded is heterogeneous in the population. Previous studies have suggested that it is not possible to reliably estimate the total population size when capture heterogeneity exists. Here we approach population estimation in the presence of capture heterogeneity as a latent length biased nonparametric density estimation problem on the unit interval. We show that in this setting it is generally impossible to estimate the density on the entire unit interval in finite samples, and that estimators of the population size have high and sometimes unbounded risk when the density has significant mass near zero. As an alternative, we propose estimating the population of individuals with capture probability exceeding some threshold. We provide methods for selecting an appropriate threshold, and show that this approach results in estimators with substantially lower risk than estimators of the total population size, with correspondingly smaller uncertainty, even when the parameter of interest is the total population. The alternative paradigm is demonstrated in extensive simulation studies and an application to snowshoe hare multiple recapture data. 
James E. Johndrow, Kristian Lum, Daniel Manrique-Vallier
stat.ME