  We study the minimax optimal rates for estimating a range of Integral
Probability Metrics (IPMs) between two unknown probability measures, based on
$n$ independent samples from them. Curiously, we show that estimating the IPM
itself between probability measures, is not significantly easier than
estimating the probability measures under the IPM. We prove that the minimax
optimal rates for these two problems are multiplicatively equivalent, up to a
$\log \log (n)/\log (n)$ factor.
