Likelihood ratio confidence interval for the abundance under binomial detectability models

Binomial detectability models are often used to estimate the size or abundance of a finite population in biology, epidemiology, demography and reliability. Special cases include incompletely observed multinomial models, capture–recapture models, and distance sampling models. The most commonly-used confidence interval for the abundance is the Wald-type confidence interval, which is based on the asymptotic normality of a reasonable point estimator of the abundance. However, the Wald-type confidence interval may have poor coverage accuracy and its lower limit may be less than the number of observations. In this paper, we rigorously establish that the likelihood ratio test statistic for the abundance under the binomial detectability models follows the chisquare limiting distribution with one degree of freedom. This provides a solid theoretical justification for the use of the proposed likelihood ratio confidence interval. Our simulations indicate that in comparison to the Wald-type confidence interval, the likelihood ratio confidence interval not only has more accurate coverage rate, but also exhibits more stable performance in a variety of binomial detectability models. The proposed interval is further illustrated through analyzing three real data-sets.