In the answer of this question, a practical solution was suggested for:
Let $X_1, X_2, ..., X_k$ be a sequence of i.i.d. random variables with $X_i \sim \mathcal{U}\{1, 2, ..., n\}$ (discrete uniform distribution). The parameter $n$ is unknown. Let $U$ be the number of unique values seen in the sequence. Given $k$ and $U$, how to estimate $n$?
thanks to:
$$E(U) = n\left(1-\left(1- \frac1n\right)^k\right)$$
it "seems a good idea", given a value $u$ and a value $k$ to choose an estimator $\hat n$ of $n$ as the real root $> 1$ of the equation $x\left(1-\left(1- \frac1x\right)^k\right) - u = 0.$
How to turn this "seems a good idea" argument into a real probabilistic/statistic argument, i.e. how to create a rigourously defined estimator (using Estimation theory in Statistics) from this?
Additionally, how could we create a 95% confidence interval from this?
Note: I already read this answer and the article from W. Esty (Ann. Statist., 1983) but this needs extra information like the number of items seens once or twice, which I don't want to use.
