  The statistical problem of using an initial sample to estimate the number of
species in a larger sample has found important applications in fields far
removed from ecology. Here we address the general problem of estimating the
number of species that will be represented by at least a number r of
observations in a future sample. The number r indicates species with sufficient
observations, which are commonly used as a necessary condition for any robust
statistical inference. We derive a procedure to construct consistent estimators
that apply universally for a given population: once constructed, they can be
evaluated as a simple function of r. Our approach is based on a relation
between the number of species represented at least r times and the higher
derivatives of the expected number of species discovered per unit of time.
Combining this relation with a rational function approximation, we propose
nonparametric estimators that are accurate for both large values of r and
long-range extrapolations. We further show that our estimators retain
asymptotic behaviors that are essential for applications on large-scale
datasets. We evaluate the performance of this approach by both simulation and
real data applications for inferences of the vocabulary of Shakespeare and
Dickens, the topology of a Twitter social network, and molecular diversity in
DNA sequencing data.
