Comparing Two Independent Groups Via a Quantile Generalization of the Wilcoxon-Mann-Whitney Test

Abstract

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: ξ_0.5, where ξ_0.5 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, ξ_q + ξ_1-q = 0 for any q ≤ 0.5, where ξ_q is the q^th quantile. Methods aimed at testing H₀: p = 0.5 are generalized by suggesting a method for testing H₀: ξ_q + ξ_1-q = 0, q < 0.5