Negative binomial information in capture-recapture sampling ($\hat{p}=\frac{r}{r+\bar{x}},I_n(p)=\frac{nr}{p^2(1-p)}$)

When each observation counts untagged captures before the $r$th tagged capture with immediate replacement, its negative binomial distribution has success probability $p=k/N$. For $n$ independent runs, the log-likelihood is $nr\log p+({\sum }_i{x}_i)\log (1-p)$ plus a constant. Differentiation gives the displayed estimator and expected Fisher information. For an interior true parameter, its asymptotic variance is $p^2(1-p)/(nr)$. A confidence interval $(L,U)$ for $p$ transforms to $(k/U,k/L)$ for the population size, with endpoint order reversed.