Possible Duplicate:
Derive househould weights from a uniformly distributed person sample
EDIT: Essentially, I have answered the question myself in the linked question: Derive househould weights from a uniformly distributed person sample. I am keeping the text below for reference.
For a finite population $U$, I know the probability $\pi_{i,k}$ of sampling case $i$ exactly $k$ times, where $k$ is limited by $m_i$. In my case,
$$\pi_{i,k} = \binom{m_i}{k} p^k(1-p)^{m_i - k}$$
for some $p$. I also have a sample of size $n$ (taken with replacement), but the total size $N$ of the population is unknown.
I am trying to derive a reweighting $w_i$ of the sample that will give unbiased and/or minimum variance estimates of totals for, say, an attribute $x_i$. (Estimating $N$ is a special case.) What would be the correct weights for my sample? Is it advisable to remove the duplicates, given that I can reliably identify them? EDIT: What if duplicates cannot be reliably identified?
My initial guess is $w_i = 1/(1 - \pi_{i, 0})$, but I don't see how to back this with mathematical argumentation. Also, I'd appreciate any hints on literature concerning this topic. (Perhaps this very problem has been treated in a previous paper?)
See Derive househould weights from a uniformly distributed person sample for a more practical description of the problem. The question asked here is different.
Perhaps Calculating % unsampled in sampling with replacement is related.
