  Estimation of the number of species or unobserved classes from a random
sample of the underlying population is a ubiquitous problem in statistics. In
classical settings, the size of the sample is usually small. New technologies
such as high-throughput DNA sequencing have allowed for the sampling of
extremely large and heterogeneous populations at scales not previously
attainable or even considered. New algorithms are required that take advantage
of the size of the data to account for heterogeneity, but are also sufficiently
fast and scale well with large data. We present a non-parametric moment-based
estimator that is both computationally efficient and is sufficiently flexible
to account for heterogeneity in the abundances of underlying population. This
estimator is based on an extension of a popular moment-based lower bound (Chao,
1984), originally developed by Harris (1959) but unattainable due to the lack
of economical algorithms to solve the system of nonlinear equation required for
estimation. We apply results from the classical moment problem to show that
solutions can be obtained efficiently, allowing for estimators that are
simultaneously conservative and use more information. This is critical for
modern genomic applications, where there may be many large experiments that
require the application of species estimation. We present applications of our
estimator to estimating T-Cell receptor repertoire and dropout in single cell
RNA-seq experiments.
