A note on marginal count distributions for diversity estimation

Abstract

Our problem is to estimate the total number of classes in a population, both observed and unobserved. This is often called the species problem, where the classes are (biological) species, but the same methods apply to “single source” capturerecapture, where only the number of captures for each individual is available (as opposed to the complete capture history). The data is summarized by the frequency counts, i.e., the number of classes observed exactly once, twice, three times, and so on, in the sample. Almost every known statistical procedure uses a mixed Poisson distribution to model the frequency counts, which assumes that the class sizes were independently generated from some latent or underlyingmixing distribution, and that the classes independently contributed members to the sample. To depart from these assumptions we require different marginal distributions for the frequency counts. Here we consider distributions having probability generating functions based on generalized hypergeometric functions, first proposed by Kemp in 1968. We show that many of these are not mixed Poisson, and are useful and valuable in the species problem.