The Wilcoxon-Mann-Whitney test, as well as modern improvements, are based in part on an estimate of p = P(D < 0), where D = X−Y and X and Y are independent random variables; a common goal is to test H0: p = 0.5. This corresponds to testing H0: ξ0.5, where ξ0.5 is the 0.5 quantile of the distribution of D. If the distributions associated with X and Y do not differ, then D has a symmetric distribution about zero. In particular, ξq + ξ1-q = 0 for any q ≤ 0.5, where ξq is the qth quantile. Methods aimed at testing H0: p = 0.5 are generalized by suggesting a method for testing H0: ξq + ξ1-q = 0, q < 0.5
Wilcox, Rand R.
"Comparing Two Independent Groups Via a Quantile Generalization of the Wilcoxon-Mann-Whitney Test,"
Journal of Modern Applied Statistical Methods:
2, Article 2.
Available at: http://digitalcommons.wayne.edu/jmasm/vol11/iss2/2