Estimation of the number of classes

A population is partitioned into S classes, and a sample of N individuals is drawn. The number of classes in which no individuals appear is unobservable, and the estimation of this quantity, or alternatively the total number of classes S, has been considered by many authors. Sichel (1986a, 1986b) considered the problem in the context of word frequency distributions and the estimation of an author's vocabulary. Using a mixed Poisson model for species abundance, Sichel (1992) proposed an estimator for the total number of species, which was also arrived at by Ord and Whitmore (1986). In this paper we show that these estimators are equivalent under maximum likelihood estimation, give a method for maximum likelihood estimation, derive the asymptotic variances of the parameter estimators and Ŝ, and investigate by simulation the sampling distribution of Ŝ.