How to determine margin of error for a population size estimation using a size-biased sampling process?

Question

I am trying to estimate the size of a discrete population $\Omega$. To do so, I draw independent samples $s_1\ldots s_n\in\Omega$ according to some distribution $p(s):\Omega\to(0,1)$. For each sample $s$, the probability $p(s)$ of having drawn $s$ is known exactly.

To estimate the size of $\Omega$, I treat $s_1,\ldots,s_n$ as size-biases samples $p(s_1),\ldots,p(s_n)$ biased by themselves. Thus, the expected value $\mathop{{}\mathrm E^*}\bigl[p^{-1}(s)\bigr]$ of the biased sample is equal to the reciprocal expected value $\mathop{{}\mathrm E}[p(s)]^{-1}$ of an unbiased sample. This value is equal to the size of $\Omega$.

I've confirmed that this method works fine and yields correct results. How can I determine variance and margin of error for the estimated size of $\Omega$?

Answer 1

I've ended up using the standard approach of computing the sampling error by modelling $E[p(s)]^{-1}$ to be normal distributed with standard deviation $s=\sqrt{\sigma^2/n}$ where $\sigma$ is the standard deviation of $p(s)^{-1}$. These error figure seem to match the kind of sampling error I'm getting, but I'm still not entirely sure if this is correct.