# Estimating the size of a sub-sample of a known population distribution given only the max and min values of the sub-sample

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 $\max X_i$?

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Might this be related to the German Tank Problem? –  Max Aug 12 '12 at 0:36
@Max, thanks for the reference but I think that problem (using the way I've framed my question) is concerned with the case where $n$ is unknown and needs to be estimated. I'm interested in the case where $n$ is known but the size of the sample, $m$, is not. –  Wiener.Snausage Aug 12 '12 at 0:52
My apologies. I was confused. Out of curiosity, what's motivating this question? Also, have you considered doing some maximum likelihood estimation based on order statistics or simulation? –  Max Aug 12 '12 at 1:18
@Max, actually this is a simplified version of something that came up while playing in a poker tournament online. The site I was playing on doesn't tell you exactly how many people are left. But, it does list the tables, sorted by table number. It is hard to count the number of tables since there may be hundreds of tables and it refreshes itself as you scroll through. It is easy to see the max and min number, though. I'm making the simplifying assumption that there are the same number of people at each table (this is usually true) and the table number is independent of the play at the table. –  Wiener.Snausage Aug 12 '12 at 4:34
@Max, so my goal is a quick way to figure out how many people are left without counting them manually. I'm just looking for a good estimator that is only a function of the max and min. Maximum likelihood estimation is a possibility but I'm having a bit of trouble deriving the joint likelihood of the max and the min, as a function of $m$. Any assistance would be appreciated. –  Wiener.Snausage Aug 12 '12 at 4:36

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.
I think the distribution of the $X_i$ has been given - it is the $i$'th selection from a sequence of equal probability samples without replacement from the list of numbers $1,2,...,n$. My question is, if all I know are the max and the min number drawn, can I estimate the number of samples that were drawn? –  Wiener.Snausage Aug 12 '12 at 0:04
are you sure there would be no information? It seems that if max = n and min = 1, this makes it more likely that $m$ is a larger number, i.e. closer to n. Similarly, if max=min, then you know $m=1$, or max=min+1, then you know $m=2$, etc. If $n$ were much larger than $m$, these kind of obvious cases aren't likely to arise but in general I'm thinking some kind of reasonable estimator is possible. –  Wiener.Snausage Aug 12 '12 at 1:01