A General Theory of the Sampling Process with Applications to the “Veil Line”

When a community of species is sampled, nonappearing species are not those with abundances that fall shy of some arbitrary mark, the “veil line” proposed by E. F. Preston in 1948 (Ecology29, 254–283). Instead, they follow a hypergeometric distribution, which has no resemblance to the veil line. There is therefore no justification for the truncation of distributions proposed to describe the abundances of species in natural communities. The mistake of the veil line points to the need for a general theory of sampling. If a community has a distributiongof species abundances and if samples taken of the community tend to follow distributionf, what is the relationship offtog? The seeds of such a theory are available in the work of E. C. Pielou. Using the Poisson distribution as a close approximation to the hypergeometric, one may immediately write and (in most cases) solve the transformation fromgtof. The transformation appears to preserve distribution formulas to within constants and parameters, providing yet another reason to rule out the use of truncation. Well beyond this application, the theory provides a foundation for rethinking the sampling process and its implications for ecology.