Mean and variance of ranks Consider rank data 1 to n with two groups, n=n1+n2, how would one test the null that the two groups have equal rank distributions using MOMENTS? (Wilcoxon is not the answer)
Is MLE possible to do the above?
If the ranks are for time periods, how does that affect the mean and variance.
 A: The question could be clearer -- in particular, the way it's phrased, there are any number of things we could test for when assessing equality of rank distributions using moments - are we comparing the first moment, the first two moments, the first three, ... the first $n$ moments?
Let's just look at the first moment to start:
The ranks of $n$ items without ties are the numbers $1,2,\ldots,n$
Under the null, $n_1$ ranks are selected randomly from those $n$ to be the ranks of the first group and $n_2$ ranks are selected randomly to form the ranks of the second group.
So under the null the average rank in the first group will be $\frac{n+1}{2}$ and so will
the average rank in the second group.
If we take the difference in the average ranks in the sample we get a statistic that would be a reasonable test statistic for equality of rank distributions.
But that's an equivalent test to the using sum of ranks in say the first sample (they are linear transformations of each other with constant coefficients at given sample sizes). Which is to say, if you're comparing first moments, you are doing a Wilcoxon test.
Note that the variance of ranks is also not so hard. If you have a random selection of $n_1$ values (without replacement) from the set $1,2,...,n$, then the average squared rank in the first sample is fairly easy to work out, and similarly the variance of the mean rank in the first sample, and so on.
You could of course construct some statistic that looked at second moments, or first and second moments or some function of the first three moments and so on.

If the ranks are for time periods, how does that affect the mean and variance.

It depends on what kind of dependence (if any) there is; it might make no difference; it might affect the variance depending on the particular kind of dependence.
