Suppose you have $n$ objects labeled $1, 2, ..., n$ and $m$ of these objects are chosen with equal probability without replacement. The labels for each of the samples are denoted by $X_1, ..., X_m$. What are some effective ways to estimate $m$ if all you observe are $\min X_i$ and $\max X_i$?
I don't think there is any effective way to do this without knowing something about the distribution of the X$_i$. But if for example the you know that the X$_i$ have an absolutely continuous distribution then you know max X$_i$ increases to b and the min X$_i$ decrease to a.
But it takes more to estimate m. Let's assume X$_i$ has a uniform dsitribution. Then E[Max X$_i$]-E[Min X$_i$] is a function of the sample size and b-a. So m can be estimated by comparing Max X$_i$-Min X$_i$ to b-a.
Now that I see that the distribution is uniform on the integers from 1 to n something can be said. You can calculate E[max X$_i$] and E[min X$_i$] as a function of m and n and use the value of Max Xi and Min X$_j$ to estimate m. However it is possible that Max X$_i$ =n and Min X$_j$=1 in which case you have nothing to help you estimate m.