Inference on the number of species through geometric lower bounds

Estimating the number of species in a population from a sample of individuals is investigated in a nonparametric Poisson mixture model. A sequence of lower bounds to the odds that a species is unseen in the sample are proposed from a geometric perspective. A lower bound and its representing mixing distribution can be computed by linear programming with guaranteed convergence. These lower bounds can be estimated by the maximum likelihood method and used to construct lower confidence limits for the number of species by the bootstrap method. Computing the nonparametric maximum likelihood estimator is discussed. Simulation is used to assess the performance of estimated lower bounds and compare them with several existing estimators. A genornic application is investigated.