Application of Laplace's boundary-mode approximations to estimate species and shared species richness

The Laplace method for approximating integrals is a useful technique in a number of research fields. This paper shows that it also has interesting applications in biological and ecological statistical inferences. When sample abundance or replicated incidence (i.e., presence or absence) records of each species are available, the expected low-order frequency counts in heterogeneous communities can be approximated by the Laplace method when the species discovery or detection probabilities are bounded from below by a constant. The approximation formulae as applied to one community can then be used to derive estimators of species richness and to examine their performance. The approach is also extended to obtain simple and new estimators for the number of shared species in two communities. The replicated species incidence data recorded by competing teams of the Hong Kong Big Bird Race for the years 1999 and 2000 are used to estimate the number of resident birds in Hong Kong and to illustrate the method of estimation.