  Given samples from two non-negative random variables, we propose a family of
tests for the null hypothesis that one random variable stochastically dominates
the other at the second order. Test statistics are obtained as functionals of
the difference between the identity and the Lorenz P-P plot, defined as the
composition between the inverse unscaled Lorenz curve of one distribution and
the unscaled Lorenz curve of the other. We determine upper bounds for such test
statistics under the null hypothesis and derive their limit distribution, to be
approximated via bootstrap procedures. We then establish the asymptotic
validity of the tests under relatively mild conditions and investigate finite
sample properties through simulations. The results show that our testing
approach can be a valid alternative to classic methods based on the difference
of the integrals of the cumulative distribution functions, which require
bounded support and struggle to detect departures from the null in some cases.
