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Suppose you have an urn filled with money. You know there are $N$ total bills, and that each is either a \$1, \$10, or \$100. You draw $n$ bills without replacement from the urn, and wish to construct, say, a confidence lower bound for the total amount of money in the urn.

The obvious solution is if $n$ is sufficiently large, then you can use the normal approximation - but let's assume that $n$ is small and/or the distribution of bills is highly skewed, so that we don't expect anything resembling decent coverage from CLT-based confidence intervals. Are there any effective exact methods for computing confidence bounds for these sorts of problems?

If you have only two types of bills, then it's not difficult (e.g. Buonaccorsi). If you know that each bill is either a \$1 or \$10, and your sample returns $k$ \$10s and ($n-k$) \$1s, then an $\alpha$-confidence lower bound for the total number of \$10s is the smallest $K$ satisfying:

$P$(draw at least $k$ out of $n$ \$10s | $K$ out of $N$ are \$10s) > $\alpha$.

Of course, in this case, having a lower bound for the total number of $10s is equivalent to having a lower bound for the total amount of money - but this no longer holds when you have more than two types of bills.

Tromb
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  • This trinomial problem is very closely related to the one discussed at http://stats.stackexchange.com/questions/18603. There, the "bills" have values $-1$, $0$, and $1$, but otherwise the issue is the same and the model is the same; the only real difference is that the present sample is without replacement from a finite population. – whuber Jul 11 '14 at 15:21
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    Yup, these are very similar problems - I'd be almost as happy if there's an answer (i.e. a way to obtain exact confidence intervals) in the multinomial case, as what I'm working with will generally have $n$ very small compared to $N$ anyway, – Tromb Jul 11 '14 at 15:39

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