  Multiple systems estimation strategies have recently been applied to quantify
hard-to-reach populations, particularly when estimating the number of victims
of human trafficking and modern slavery. In such contexts, it is not uncommon
to see sparse or even no overlap between some of the lists on which the
estimates are based. These create difficulties in model fitting and selection,
and we develop inference procedures to address these challenges. The approach
is based on Poisson log-linear regression modeling. Issues investigated in
detail include taking proper account of data sparsity in the estimation
procedure, as well as the existence and identifiability of maximum likelihood
estimates. A stepwise method for choosing the most suitable parameters is
developed, together with a bootstrap approach to finding confidence intervals
for the total population size. We apply the strategy to two empirical data sets
of trafficking in US regions, and find that the approach results in stable,
reasonable estimates. An accompanying R software implementation has been made
publicly available.
