Capture-recapture is a method commonly used in ecology to estimate an animal population's size. Let
$N$ = Number of animals in the population;
$K$ = Number of animals marked on the first visit;
$n$ = Number of animals captured on the second visit;
$k$ = Number of recaptured animals that were marked.
It is assumed that all individuals have the same probability of being captured in the second sample, regardless of whether they were previously captured in the first sample.
This implies that, in the second sample, the proportion of marked individuals that are caught ($k/K$) should equal the proportion of the total population that is caught ($n/N$).
In symbols, $\frac{k}{K} = \frac{n}{N}$. This derives the Lincoln–Petersen estimator$$\hat{N}_{LP}=\frac{Kn}{k}$$
The problem is how to calculate the variance of $\hat{N}_{LP}$.
I have tried the hypergeometric distribution likelihood, and treated the sample size $K,n$ as fixed. This model contains one parameter $N$ and one random variable $k$. I can get that $\hat{N}_{LP}$ is the MLE of $N$, but get stucked with its variance.
Thanks for your help.
Best Answer
Before addressing the variance, one should note that the Lincoln-Petersen estimator is not well defined with full probability since, unless $K+n\geqslant N+1$ (a highly unrealistic hypothesis), the event $k=0$ happens with positive probability.
To alleviate this problem, one rather considers the Chapman estimator $$\hat N_C=\frac{(K+1)(n+1)}{k+1}-1,$$ whose variance is $$ \operatorname{var}(\hat N_C) = \frac{(K+1)(n+1)(K-k)(n-k)}{(k+1)^2(k+2)}.$$ See the obvious for some references.