Some General Comparative Points on Chao's and Zelterman's Estimators of the Population Size

Abstract. Two simple and frequently used capture–recapture estimates of the population size are compared: Chao's lower‐bound estimate and Zelterman's estimate allowing for contaminated distributions. In the Poisson case it is shown that if there are only counts of ones and twos, the estimator of Zelterman is always bounded above by Chao's estimator. If counts larger than two exist, the estimator of Zelterman is becoming larger than that of Chao's, if only the ratio of the frequencies of counts of twos and ones is small enough. A similar analysis is provided for the binomial case. For a two‐component mixture of Poisson distributions the asymptotic bias of both estimators is derived and it is shown that the Zelterman estimator can experience large overestimation bias. A modified Zelterman estimator is suggested and also the bias‐corrected version of Chao's estimator is considered. All four estimators are compared in a simulation study.