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

• Published in: Journal of the American Statistical Association Vol. 93; no. 441; pp. 372 - 379
• Main Authors: Boneh, Shahar, Boneh, Arnon, Caron, R. J.
• Format: Journal Article
• Language: English
• Published: Alexandria, VA Taylor & Francis Group 01-03-1998; American Statistical Association; Taylor & Francis Ltd
• Subjects: Algorithms; Alternating copositivity; Coverage; Detection; Estimating population size; Estimation; Estimation bias; Estimators; Exact sciences and technology; Forecasts; Inference from stochastic processes; time series analysis; Mathematical functions; Mathematical programming; Mathematics; Modeling; Parametric inference; Poisson process; Polynomials; Population; Predictions; Probability and statistics; Probability theory and stochastic processes; Sampling; Sciences and techniques of general use; Species; Statistical analysis; Statistical methods; Statistical variance; Statistics; Stochastic processes; Theory and Methods; Prediction theory; Numerical simulation; Sample size; Sampling
• ISSN: 0162-1459; 1537-274X
• DOI: 10.1080/01621459.1998.10474118

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 and Thisted lacks some essential properties of the true prediction function, for example, the property of alternating copositivity. As a result it cannot be used for large values of t. We propose an alternative estimator that possesses the essential properties and is easily obtained. We illustrate our estimator with two numerical examples and a simulation study.