Say that I have a urn with, $m$, unique samples. There is a probability, $p_{i}$ for sampling a given unique sample, $m_{i}$, when sampling with replacement. Also note that the probability, $p_{i}$, of sampling a unique sample, $m_{i}$, is not necessarily uniform among all samples within in $m$. I know that the general formulation for determining the expected number of samples for a coupon collector problem is:
$$ E[T]=\int_0^\infty \left(1-\prod_{j=1}^m (1-e^{-p_j t})\right) dt, $$
I also know that to find the expected number of samples required to sample a subset, $s$, of $m$ is (equation 45 here: https://pdfs.semanticscholar.org/077f/29af3a2e01658a467debf479d41288439a93.pdf):
$$ E[T]=\int_0^\infty \left(1-\prod_{j\in s} (1-e^{-p_j t})\right) dt, $$
My question is how do you find the expected number of samples required to sample a given fraction of $m$? That is, I do not necessarily care about sampling a specific subset, $s$, but only that some unique fraction of the population $m$ is represented (e.g., half of the unique $m$ are sampled). In addition, if anyone has a reference source, I would greatly appreciate it. Thank you.
This is part of the so-called unseen species problem, and you can find a Bayesian solution in this paper.