Transformed Logit Confidence Intervals for Small Populations in Single Capture–Recapture Estimation

Abstract

The good performance of logit confidence intervals for the odds ratio with small samples is well known. This is true unless the actual odds ratio is very large. In single capture–recapture estimation the odds ratio is equal to 1 because of the assumption of independence of the samples. Consequently, a transformation of the logit confidence intervals for the odds ratio is proposed in order to estimate the size of a closed population under single capture–recapture estimation. It is found that the transformed logit interval, after adding .5 to each observed count before computation, has actual coverage probabilities near to the nominal level even for small populations and even for capture probabilities near to 0 or 1, which is not guaranteed for the other capture–recapture confidence intervals proposed in statistical literature. Thus, given that the .5 transformed logit interval is very simple to compute and has a good performance, it is appropriate to be implemented by most users of the single capture–recapture method.

Keywords:

• Asymptotic normality
• Coverage probability
• Dual-system estimation
• Expected width
• Monte Carlo study
• Odds ratio
• Pivotal statistic
• Profile likelihood

Mathematics Subject Classification:

• 62F25
• 62P12

Acknowledgments

The author thanks to B. P. Urdinola, L. M. González, S. T. Buckland, A. Agresti, and an anonymous reviewer for reading the preliminary versions of this work and their helpful comments and suggestions. Also, to C. E. Pardo, A. Irlande, S. Baillargeon, and L.-P. Rivest for their help about programing the simulations and computations in this work.