Capture–Recapture Estimation by Means of Empirical Bayesian Smoothing with an Application to the Geographical Distribution of Hidden Scrapie in Great Britain

Abstract

Summary The paper discusses population size estimation on the basis of a frequency distribution of zero-truncated counts and is motivated by a study on the geographical distribution of hidden scrapie in Great Britain. Aggregation of scrapie cases is considered at the county level and results in sparse zero-truncated count distributions which make the application of conventional capture–recapture procedures for estimating the hidden part of the scrapie-affected population difficult. We suggest a smoothed generalization of Zelterman’s estimator of population size which overcomes the overestimation bias of the conventional Zelterman estimator and instead produces a lower bound, which is typically larger than Chao’s lower bound estimator. The estimator uses an empirical Bayes approach with various choices for the prior distribution including a parametric choice of the gamma distribution as well as various non-parametric distributions. A simulation study investigates the performance of the new estimators, and also in comparison with conventional estimators. The empirical Bayes estimator with a non-parametric mixture model as prior performs well and the boundary problem of the conventional non-parametric discrete mixture model estimator leading to spurious population size is avoided. In the application to hidden scrapie in Great Britain the new estimators lead to maps of scrapie of observed–hidden ratios as well as completeness of the current surveillance system.