Estimating the Prediction Function and the Number of Unseen Species in Sampling with Replacement

A sample of N units is taken from a population consisting of an unknown number of species. We are interested in estimating the number of species and the prediction function for future sampling. The prediction function is defined as the expected number of new species that will be found if an additional sample of size tN is taken, for any positive real number t. In this paper we point out that an estimator suggested by Efron & Thisted (1976) lack some essential properties of the true prediction function, e.g., the property of alternating copositivity. As a result, it cannot be used for large values of t. We propose an alternative estimator which possesses the essential properties, and is easily obtained. We illustrate our estimator with two numerical examples and a simulation study.