Five Confidence Intervals of the Closed Population Size in the Capture-recapture Problem under Inverse Sampling with Replacement

In the capture-recapture problem for two independent samples, the traditional estimator, calculated as the product of the two sample sizes divided by the number of sampled subjects appearing commonly in both samples, is well known to be a biased estimator of the population size and have no finite variance under direct or binomial sampling. To alleviate these theoretical limitations, the inverse sampling, in which we continue sampling subjects in the second sample until we obtain a desired number of marked subjects who appeared in the first sample, has been proposed elsewhere. In this paper, we consider five interval estimators of the population size, including the most commonly-used interval estimator using Wald’s statistic, the interval estimator using the logarithmic transformation, the interval estimator derived from a quadratic equation developed here, the interval estimator using the c2-approximation, and the interval estimator based on the exact negative binomial distribution. To evaluate and compare the finite sample performance of these estimators, we employ Monte Carlo simulation to calculate the coverage probability and the standardized average length of the resulting confidence intervals in a variety of situations. To study the location of these interval estimators, we calculate the non-coverage probability in the two tails of the confidence intervals. Finally, we briefly discuss the optimal sample size determination for a given precision to minimize the expected total cost.