Consider the classical capture-recapture method, where we are to estimate the number of deer (say) in a sanctuary. So a certain number of deer is captured, tagged and released. Then a random sample is drawn and the number of tagged deer is observed. Now if $X$ is the number of tagged deer in the sample, I have the hypergeometric model

$$P_N(X=x)=\frac{\binom{M}{x}\binom{N-M}{n-x}}{\binom{N}{n}}\,\,,\, \small x\in\left\{\max(0,n+M-N),\ldots,\min(n,M)\right\}$$

Here $N$ is population size, $M$ is the number of tagged deer and $n$ is the sample size. ($M,n,x$ are all known.)

Then $E_N(X)=\frac{nM}{N}\implies E_N(\frac{X}{nM})=\frac1N$.

So using an unbiased estimator of $1/N$, I take an estimator of $N$ to be $\hat N(X)=\frac{nM}{X}$ (rounded to nearest integer).

I am trying to find an estimate of the standard error of the estimator $\hat N$, for which I need an estimate of the variance of $\hat N$.

Now, $\operatorname{Var}_N(\hat N)=n^2 M^2\operatorname{Var}_N(\frac1 X)$ and $\operatorname{Var}_N(X)=\frac{nM}{N}\left(1-\frac MN\right)\frac{N-n}{N-1}$.

For the variance of $1/X$, I used this first order approximation to get

$$\operatorname{Var}_N\left(\frac1X\right)\approx \frac{N^2}{n^3 M^3}\cdot\frac{(N-M)(N-n)}{N-1}$$

Therefore, $$\operatorname{Var}_N(\hat N)\approx \frac{N^2}{nM}\cdot\frac{(N-M)(N-n)}{N-1}$$

For the estimated variance I substitute $\hat N$ for $N$ in the above expression, which simplifies to

$$\widehat{\operatorname{Var}_N(\hat N)}\approx \frac{n^2M^2(n-x)(M-x)}{x^3(nM-x)}$$

Assuming my calculations are correct, can this expression be reasonably approximated further? If I use the second order approximation for the variance, the calculation gets more cumbersome.

I have the expression $\widehat{\operatorname{Var}_N(\hat N)}\approx \frac{M^2n(n-x)}{x^3}$ in my notes, but without any derivation.

In Johnson-Kotz's Univariate Discrete Distributions, they mentioned the related Chapman estimator (this book was cited which could have an answer) for which several estimates of its variance have been suggested. But I could not find a reference for any standard expression of the estimated variance/standard error of the estimator $\hat N$ in particular.


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