  The energy distance and energy scores became important tools in multivariate
statistics and multivariate probabilistic forecasting in recent years. They are
both based on the expected distance of two independent samples. In this paper
we study dependence uncertainty bounds for these quantities under the
assumption that we know the marginals but do not know the dependence structure.
We find some interesting sharp analytic bounds, where one of them is obtained
for an unusual spherically symmetric copula. These results should help to
better understand the sensitivity of these measures to misspecifications in the
copula.
