  We wish to estimate the total number of classes in a population based on
sample counts, especially in the presence of high latent diversity. Drawing on
probability theory that characterizes distributions on the integers by ratios
of consecutive probabilities, we construct a nonlinear regression model for the
ratios of consecutive frequency counts. This allows us to predict the
unobserved count and hence estimate the total diversity. We believe that this
is the first approach to depart from the classical mixed Poisson model in this
problem. Our method is geometrically intuitive and yields good fits to data
with reasonable standard errors. It is especially well-suited to analyzing high
diversity datasets derived from next-generation sequencing in microbial
ecology. We demonstrate the method's performance in this context and via
simulation, and we present a dataset for which our method outperforms all
competitors.
