I am trying to understand what can be proved about minimum variance estimators. I am a little confused by Cramér–Rao and how to apply it even to really simple examples or if it's even the right tool in this case.

Let us assume we have some finite set $S$ of elements, and we just want to estimate the cardinality $n$ of $S$. We know an upper bound $N$ for the cardinality. One method is to sample with replacement, count the number of distinct elements, and form one's estimate from this using the fact that $E(\text{number of distinct elements in sample})= n(1-(1-1/n)^x)$, where $x$ is the size of the sample. I don't know if this a good method however. Another would be simple capture-recapture taking two sample sets and looking at their intersection, but this seems an odd thing do by computer.

How do we show a tight lower bound for the variance of any estimator for this simple problem? How do we prove that an unbiased estimator has the minimum variance, if indeed it does? Is the sampling method I describe optimal?

I asked a similar but more specific question at Mathematics: Fisher Information and minimum variance estimators.


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