Estimating the Size of a Population from a Single Sample

SUMMARY Methods for estimating the size of a population of individuals usually require multiple samples from the group. We consider a population composed of an unknown number, N*, of individuals on one or more of K > 1 ordered lists. A single sample of individuals from the population, those on list K, together with the identification of the list on which they last appeared prior to list K is obtained. Under relatively weak assumptions on the probability model, an unbiased maximum likelihood estimator of N* is obtained. An expression is derived for the bias of the estimator and its consequence on the true probability of coverage of the confidence interval when the model's assumptions do not hold. Applications of this method are discussed and an illustrative example is presented. Estimation of the size of a closed population is frequently based on a capture-recapture (CR) model. The data may be viewed as K > 1 samples or lists of individuals who are tagged or otherwise identified so that the lists on which they appear are known. The capture history of each individual in the population corresponds to a K-dimensional vector whose ith element is 1 if the individual is on the ith list or sampling occasion and 0 otherwise. An unobserved individual is represented by a vector all of whose elements are 0. In the CR setting the size of the population, the parameter to be estimated, is the observed number of distinct individuals appearing on at least one of the lists plus the unobserved number who do not appear on any of the lists. Viewing the observations as falling into a 2K multidimensional cross-classification, Bishop, Fienberg, and Holland (1975) reviewed the use of log-linear models for estimating the size of the population. Recently, Wolter (1986) proposed various two-dimensional cross-tabulation coverage models for census data. The literature appears to be silent on the problem of estimating the number of individuals, N*, on K lists when only one of the lists, say the last one, is observable. The individuals on this list in the CR setting are those caught in the last capture. The problem arises in many areas. For example, suppose an unknown number of patients are treated in a clinic during K = 52 weeks. In the last week of the 52-week period a survey of those who are treated is to be performed. Can information on the surveyed individuals be obtained that would permit estimation of the distinct number of patients treated in the year? Neither the standard capture-recapture models of Seber (1982), nor the contingency table approaches