A combination of Wilcoxon's and Ansari-Bradley's statistics

TLDR: In this article, a nonparametric two-sample test for location and dispersion is proposed, where two samples, the X-sample and the Y-sample of independent observations from populations with continuous distribution functions, F(x) and G(y) respectively, are con- sidered.

Abstract: SUMMARY This paper deals with a nonparametric two-sample test for location and dispersion. It Two samples, the X-sample and the Y-sample of m and n independent observations from populations with continuous distribution functions, F(x) and G(y) respectively, are con- sidered. The problem is to test the hypothesis G(x) _ F(x) versus alternatives of the form G(x) = F(ax+b) with a * 1, or b $ 0(a > 0). The proposed T statistic is a certain function of the Wilcoxon and Ansari-Bradley statistics. The Wilcoxon statistic is well known and an important property is that the two- sided test based on this statistic is consistent for testing the hypothesis F(x) = G(x) versus alternatives of the form G(x) = F(x + b) (b * 0); see Mann & Whitney (1947) and van Dant- zig (1951). The Ansari-Bradley statistic is defined as follows: in the combined samples, the observations less than or equal to the median are replaced by their ranks in increasing order and those larger than the median are replaced by their ranks in decreasing order; the statistic is the sum of these ranks for the X-sample. The two-sided test based on this statistic is consistent for testing the hypothesis F(x) = G(x) versus alternatives of the form