I'm afraid I'm not an expert in statistics, but I have a particular problem I'm interested in solving. I'm pretty sure this area already has a lot of literature, but I'm having difficulty finding something directly applicable to what I'm doing, so it would be great if somebody could nudge me in the right direction.
I'll give an example to illustrate the problem: Say I have a machine which produces biased coins. It has some underlying continuous probability density function which it is using to pick a number 0 < p < 1, then creates a coin which will come up heads with probability p.
My task is to estimate the function which the machine uses to generate these coins. I'm allowed to flip the coins, and I have access to a very large number of coins, but each one after a certain, random, amount of flips is taken away from me. The number of flips is not necessarily large.
How would I go about doing this? My initial thought was to just sum each resulting binomial distribution and divide by the total number of coins tested. But I'm pretty sure this doesn't give a good result.
I'm faintly aware of kernal density estimation, but I don't have enough expertise to know if/how it can be used for this kind of task, or what I should know in terms of tailoring it to this task.