  The theoretical advances on the properties of scoring rules over the past
decades have broadened the use of scoring rules in probabilistic forecasting.
In meteorological forecasting, statistical postprocessing techniques are
essential to improve the forecasts made by deterministic physical models.
Numerous state-of-the-art statistical postprocessing techniques are based on
distributional regression evaluated with the Continuous Ranked Probability
Score (CRPS). However, theoretical properties of such evaluation with the CRPS
have solely considered the unconditional framework (i.e. without covariates)
and infinite sample sizes. We extend these results and study the rate of
convergence in terms of CRPS of distributional regression methods. We find the
optimal minimax rate of convergence for a given class of distributions and show
that the k-nearest neighbor method and the kernel method reach this optimal
minimax rate.
