Generalized Species Richness Indices for Diversity

A generalized notion of species richness is introduced. The generalization embeds the popular index of species richness on the boundary of a family of diversity indices each of which is the number of species in the community after a small proportion of individuals belonging to the least minorities is trimmed. It is established that the generalized species richness indices satisfy a weak version of the usual axioms for diversity indices, are qualitatively robust against small perturbations in the underlying distribution, and are collectively complete with respect to all information of diversity. In addition to a natural plug-in estimator of the generalized species richness, a bias-adjusted estimator is proposed, and its statistical reliability is gauged via bootstrapping. Finally an ecological example and supportive simulation results are given.