= Negative binomial information in capture-recapture sampling
{title2=$\widehat p=\frac r{r+\bar x},\quad I_n(p)=\frac{nr}{p^2(1-p)}$}
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.
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