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Capture-recapture techniques are very powerful tool and widely used for estimating an elusive target population size. Capture-recapture count data is presented in form of frequencies data consisting of the frequency of units detected exactly once, twice and so on, with the unknown frequency of undetected units. Hence, the resulting distribution in this case is a zero-truncated count distribution. Basically, the binomial distribution and the Poisson distribution are considered as simple models which is corresponding to counting occasions. The exponential-Poisson mixture model has been widely used to construct population size estimator for capture-recapture data.

In fact, the target population might be heterogeneous because it has different characteristics, resulting in over or under dispersion based on the basic models. Under the Poisson distribution,  the original Turing estimator provides a good performance of estimating target population size. Additionally, the Turing-based geometric distribution with non-parametric approach was proposed (TG) for the heterogeneous population. It gives an easy way to estimate the target population size. In this work, we derived uncertainty measures for the TG estimator by considering two sources of variance (M1), and the second approach used only one source of variance (M2). It is emphasised that although the analytic approaches to compute uncertainty measures can be easily used in practice, there are valid asymptotically and requires a large sample size. Therefore, re-sampling approaches, true bootstraps (M3), imputed bootstrap (M4) and reduced bootstrap (M5), are proposed as alternative methods to get uncertainty measures. The study compares performance of variance and confidence intervals of paralytics and re-sampling methods by using a simulation study. Overall, the imputed bootstrap is the best choice for estimating variance and constructing confidence interval for TG estimator. The analytic approach with two sources of variance remains successful to estimate variance and calculate confidence intervals for large sample size. It is very clear that the reduced bootstrap and the analytic approach with one  source of variance is not appropriate in all situations. For the true bootstrap, the true value of population size is often unknown in nature; therefore, it is  useless for capture-recapture study

Capture-recapture, Turing estimator, Geometric distribution