Some superpopulation models for estimating the number of population uniques

The number of the unique individuals in the population is of great importance in evaluating the disclosure risk of a microdata set. We approach this problem by considering some basic superpopulation models including the gamma-Poisson model of Bethlehem et al. (1990). We introduce Dirichlet-multinomial model which is closely related but more basic than the gamma-Poisson model, in the sense that binomial distribution is more basic than Poisson distribution. We also discuss the Ewens model and show that it can be obtained from the Dirichlet-multinomial model by a limiting argument similar to the law of small numbers. The multivariate Ewens distribution is a basic mathematical model used in genetics. Estimation of the number of the population uniques is particularly simple under the Ewens model. Although these models might not necessarily well t actual populations, they can be considered as basic mathematical models for our problem, as binomial and Poisson distributions are considered as basic models for count data.