Nonparametric MLE under Two Closed Capture-Recapture Models with Heterogeneity

We conduct nonparametric maximum likelihood estimation under two common heterogeneous closed population capture-recapture models. Our models specify mixture models (as did previous researchers' models) which have a common generating distribution, say F, for the capture probabilities. Using Lindsay and Roeder's (1992, Journal of the American Statistical Association 87, 785-794) mixture model results and the EM algorithm, a nonparametric maximum likelihood estimator (MLE) of F for any specified population size N is obtained. Then, the nonparametric MLE of the (N, F) pair and thus for N is determined. Perhaps most importantly, since our MLE pair maximizes the likelihood under the entire nonparametric probability model, it provides an excellent foundation for estimating properties of estimators, conducting a goodness-of-fit test, and performing a likelihood ratio test. These are illustrated in the paper.