  A fundamental functional in nonparametric statistics is the Mann-Whitney
functional ${\theta} = P (X < Y )$ , which constitutes the basis for the most
popular nonparametric procedures. The functional ${\theta}$ measures a location
or stochastic tendency effect between two distributions. A limitation of
${\theta}$ is its inability to capture scale differences. If differences of
this nature are to be detected, specific tests for scale or omnibus tests need
to be employed. However, the latter often suffer from low power, and they do
not yield interpretable effect measures. In this manuscript, we extend
${\theta}$ by additionally incorporating the recently introduced distribution
overlap index (nonparametric dispersion measure) $I_2$ that can be expressed in
terms of the quantile process. We derive the joint asymptotic distribution of
the respective estimators of ${\theta}$ and $I_2$ and construct confidence
regions. Extending the Wilcoxon- Mann-Whitney test, we introduce a new test
based on the joint use of these functionals. It results in much larger
consistency regions while maintaining competitive power to the rank sum test
for situations in which {\theta} alone would suffice. Compared with classical
omnibus tests, the simulated power is much improved. Additionally, the newly
proposed inference method yields effect measures whose interpretation is
surprisingly straightforward.
