A ‘polymath’ project on the Wilcoxon test?

`Polymath’ is a set of projects in massive collaborative proof of mathematical results; Terry Tao and Timothy Gowers are two of the famous mathematicians involved. There’s a new potential project on Gowers’s blog, which he describes a being related to intransitive dice. As you know, if you read this blog, (a) I prefer the term non-transitive, and (b) this means it’s about the Wilcoxon test.

The idea of the conjecture is that you define an $n$ -sided die by sampling uniformly with replacement $n$ numbers from $1, 2, 3, \dots, n$ as the numbers on the sides, with the constraint that the numbers have to add up to $n (n + 1) / 2$ . Rolling such a die $M$ times samples $M$ observations from a distribution on $1, 2, 3, d o t s, n$ with mean $(n + 1) / 2$ . You construct three such dice, $A$ , $B$ , and $C$ . Their distributions have the same mean, so the $t$ -test would have no ability to distinguish data from the three distributions, no matter how large $M$ was. But the Wilcoxon test probably would. It’s assumed, but not yet proved, that the probability of a tie according to the Wilcoxon test goes to zero as $n \to \infty$ .

The interesting conjecture is that if you see $A$ beat $B$ and $B$ beat $C$ according to the Wilcoxon test, the probability that $A$ beats $C$ goes to 1/2 as $n$ goes to infinity. That is, given equal means, the Wilcoxon test is basically as non-transitive as possible.