Mathematical Properties of Continuous Ranked Probability Score Forecasting

The theoretical advances on the properties of scoring rules over the past decades have broaden 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 minimization of the CRPS have mostly considered the unconditional framework (i.e. without covariables) and infinite sample sizes. We circumvent these limitations 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. Moreover, we show that the k-nearest neighbor method and the kernel method for the distributional regression reach the optimal rate of convergence in dimension $d\geq2$ and in any dimension, respectively.