Estimation of species richness based on species range

A geographical region, containing an unknown number of species, is partitioned into N quadrats. The range of a species is defined to be the number of quadrats in which the species is present. A random sample of n quadrats is drawn without replacement, and the species list is determined for each of the selected quadrats. Two estimators of S are proposed. The inclusion probabilities in the Horvitz-Thompson estimator involve the unknown species ranges but these ranges can be estimated to yield an “estimated” Horvitz-Thompson estimator. This estimator is biased because of the use of estimated inclusion probabilities. For the other estimator, it is shown that the expected number of species in the sample having a specified sample range r is a linear combination over R of the number SR of species in the population with population range R. Letting r vary yields a system of linear equations that can be solved to obtain estimates for the SR and for S. These estimators for SR and S are shown to be unbiased when the sample size n is sufficiently large.