On the Nonidentifiability of Population Sizes

When a nonparametric mixture model is adopted to deal with the heterogeneity among individual capture probabilities, the population size is nonidentifiable (Link, 2003, Biometrics 59, 1123-1130). Holzmann, Munk, and Zucchini (2006, Biometrics 62, 934-936) discussed the conditions under which a subfamily of mixing distributions is identifiable. Link (2006, Biometrics 92, 936-939) found that the nonidentifiability occurs across identifiable subfamilies. It is shown that there is a subfamily in which each mixing distribution is determined by its mixture, and the population size admits estimable lower bounds that can be used to construct lower confidence limits.