## Estimation of a distribution from i.i.d. sums

Here’s an estimation problem that I ran into not long ago while working on a problem in entity co-reference resolution in natural language documents.

Let be a random variable taking on values in . We are given data , where is the sum of independent draws of for . We are required to estimate the distribution of from .

For some distributions of we can use the method-of-moments. For example if , we know that the mean of is . We can therefore estimate as the sample mean, i.e., . Because of the nice additive property of the parameters for sums of i.i.d. poisson random variables, the maximum likelihood estimate also turns out be the same as .

The problem becomes more difficult when is say a six-sided die (i.e., the sample space is ) and we would like to estimate the probability of the faces . How can one obtain the maximum likelihood estimate in such a case?