I would like to know what the most powerful way of comparing two (or more) discrete distributions is.
I know that the Kolmogorov-Smirnov test could be used (if corrected for the discrete ecdfs), and/or a chi-squared test, and other summary statistics could be compared (mean/variance/skewness &c), but is there a more powerful test along the lines of the Cramér–von_Mises test? There is unlikely to be much deviation across the whole of the discrete distributions, so I'd like the test to have as much power as possible, a situation that C-vM would be best suited for if the distributions were samples from a continuous distribution.
Multiple machines generate strings of a fixed length with an added 'tail' (of an random integer number - say between 0 and 250 - of special characters). Changing the environment the machines sit in may (or may not) change the tail-length distributions.
Taking all the strings from the machines at different time points will give a time-varying distribution of special character tail lengths.
I'd like to know if we can test whether there are significant changes in the tail-length distribution over the time course.