# Characterizing uncertainty in empirical PMF of unknown discrete random distribution

Suppose a discrete random distribution $X$ which takes integer values in some small (but not binary) range, e.g. $[0, 255]$ and whose true pmf is unknown (however, the null hypothesis is that it is uniform). If I draw a large sample $\{x\}$ from this distribution, I can estimate the pmf by counting the number of $x_i$ that take each value and dividing by the total. But if I draw another large sample, I will most likely get a different estimate.

I'd like some way of characterizing the uncertainty in each point in the pmf, ideally one that can be updated as I (that is, the computer) continually draws new samples.

"Confidence intervals for empirical CDF" looks closely related to what I'm trying to do but AFAICT only applies to continuous distributions.

• I don't see where the DKW inequality is limited to continuous distributions based on the wikipedia article en.wikipedia.org/wiki/… – soakley May 7 '13 at 15:37
• The linked question made it sound like it was only good for continuous distributions, but I've now done a little more digging and it isn't: for instance Massart's "The Tight Constant in the Dvoretzky-Kiefer-Wolfowitz Inequality" is mostly over my head, but does explicitly show that the inequality is valid for discrete distributions. – zwol May 7 '13 at 16:13