Penalized likelihood inference for the finite mixture of Poisson distributions from capture-recapture data.

In capture-recapture problems, when individuals are categorized into different groups and individuals within each group are suspected to be captured with equal probability, the finite mixture of Poisson distributions is commonly employed to address heterogeneity in capture probabilities. In this study, we propose a penalized likelihood estimation method to estimate population sizes and demonstrate that the penalized likelihood ratio statistic asymptotically follows a standard chi-square distribution. To detect the presence of heterogeneity, we introduce a retooled EM test statistic that asymptotically follows a mixture of chi-square distributions. Our numerical investigations reveal that the proposed maximum penalized likelihood estimator offers increased stability, while the penalized likelihood ratio interval estimator shows enhanced accuracy compared with existing approaches. By carefully selecting an adaptive tuning parameter, the EM test achieves a better balance between the type I error and power than the goodness-of-fit and AIC-based tests. Finally, we apply the proposed method to three real-life datasets: street prostitute data, H5N1 influenza data, and opiate user data. [ABSTRACT FROM AUTHOR]