I am trying to understand what can be proved about minimum variance estimators. I have changed the question to make it more specific.
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 might be to sample with replacement and 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.
How do we compute the Fisher Information for this problem? Ultimately I would like to show a lower bound but this would be great first step.