  Distribution-on-distribution regression considers the problem of formulating
and estimating a regression relationship where both covariate and response are
probability distributions. The optimal transport distributional regression
model postulates that the conditional Fr\'echet mean of the response
distribution is linked to the covariate distribution via an optimal transport
map. We establish the minimax rate of estimation of such a regression function,
by deriving a lower-bound that matches the convergence rate attained by the
Fr\'echet least squares estimator.
