Multiple collection estimation of population size: A generalization of capture-recapture

We consider a class of statistical models generalizing the very useful, classical capture-recapture model. The classical model involves two independent surveys, and hence (usually) two collectors. Our generalizations allow for an arbitrary number of collectors, and generalize the classical model also in some other respects. The problem is precisely formulated; minimal sufficient statistics are described; and the maximum likelihood estimator is derived and its existence and uniqueness are established. Asymptotic properties of this MLE are then studies in three separate asymptotic regimes. For one of these it turns out, rather surprisingly that the MLE is not asymptotically efficient, and can be improved. This work further develops modern asymptotic theory in settings in which 1) the dimension of the sample space grows with the number of observations and 2) the data and parameter space are each discrete. A variant of the Cramer-Rao inequality is derived for such settings, and is used in our analysis of the multiple collector problem