  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.
