What I'm Working On

I'll have a bigger, more "friendly" writeup here in a while, but here's what I'm working on for now:

We have n i.i.d. geometric random variables with parameter q, but we do not know n or q. Instead, for some (known) k < n, we are given the largest k of these values. Our challenge is to estimate n and q based on this information.

I think this is a reasonable first model for a situation where we are trying to separate "talent" from "persistance" or "luck" -- i.e., though I'm terrible at basketball, I can make an arbitrarily long series of free throws if I just go and keep throwing balls at the basket over and over and over again. If all you can see are my "high scores," can you tell how good I am?

For fixed k, it's not too hard to go after this kind of problem with spreadsheets and so forth, but I'm hoping that I might be able to get some more interesting mathematical results. At this point I'm having a hard time deciding on what sort of approach to use. One is to consider the set N x [0,1] and consider the marginal probability that N samples of a Geometric(q) distribution would have generated such a set of results; the other is to try to consider the relation between the sample and population variance.

This is just a bit of a toy problem and I haven't really had too much time to work on it while I've been looking for jobs and so forth, but I think it has the potential to be somewhat interesting -- particularly in the case k = 5, which will let you rag on your friends for their performance on WiiPlay.