  A finite set is "hidden" if its elements are not directly enumerable or if
its size cannot be ascertained via a deterministic query. In public health,
epidemiology, demography, ecology and intelligence analysis, researchers have
developed a wide variety of indirect statistical approaches, under different
models for sampling and observation, for estimating the size of a hidden set.
Some methods make use of random sampling with known or estimable sampling
probabilities, and others make structural assumptions about relationships (e.g.
ordering or network information) between the elements that comprise the hidden
set. In this review, we describe models and methods for learning about the size
of a hidden finite set, with special attention to asymptotic properties of
estimators. We study the properties of these methods under two asymptotic
regimes, "infill" in which the number of fixed-size samples increases, but the
population size remains constant, and "outfill" in which the sample size and
population size grow together. Statistical properties under these two regimes
can be dramatically different.
