Nonparametric lower bounds for species richness and shared species richness under sampling without replacement

A number of species richness estimators have been developed under the model that individuals (or sampling units) are sampled with replacement. However, if sampling is done without replacement so that no sampled unit can be repeatedly observed, then the traditional estimators for sampling with replacement tend to overestimate richness for relatively high-sampling fractions (ratio of sample size to the total number of sampling units) and do not converge to the true species richness when the sampling fraction approaches one. Based on abundance data or replicated incidence data, we propose a nonparametric lower bound for species richness in a single community and also a lower bound for the number of species shared by multiple communities. Our proposed lower bounds are derived under very general sampling models. They are universally valid for all types of species abundance distributions and species detection probabilities. For abundance data, individuals' detectabilities are allowed to be heterogeneous among species. For replicated incidence data, the selected sampling units (e.g., quadrats) need not be fully censused and species can be spatially aggregated. All bounds converge correctly to the true parameters when the sampling fraction approaches one. Real data sets are used for illustration. We also test the proposed bounds by using subsamples generated from large real surveys or censuses, and their performance is compared with that of some previous estimators.