Statistical Inference in the Classical Occupancy Problem Unbiased Estimation of the Number of Classes

Abstract

A random sample of N observations is obtained from a multinomial distribution with an unknown number θ of equiprobable cells. The existence and asymptotic properties of the unbiased estimator based on the number of occupied cells is given. The related problem in which observations are made sequentially, stopping after L repeated cells have been observed, is also discussed. In the fixed sample size case, if θ is known not to exceed N, the unbiased estimator is a ratio of Stirling numbers of the second kind. For the sequential ease, the unbiased estimator is always a ratio of Stirling numbers of the second kind.