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 unknow n number of species. We are interested in estimating the number of spe cies and the prediction function for future sampling. The prediction f unction 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 rea l number t. In this paper we point out that an estimator suggested by Efron and Thisted lacks some essential properties of the true predicti on function, for example, the property of alternating copositivity. As a result it cannot be used for large values of t. We propose an alter native estimator that possesses the essential properties and is easily obtained. We illustrate our estimator with two numerical examples and a simulation study.