# Distributional regression and its evaluation with the CRPS: Bounds and convergence of the minimax risk

arXiv: https://arxiv.org/abs/2205.04360 · HTML: https://mlsanity.com/paper/2205.04360/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 27 references that are in the map.
- First posted on arXiv 9 May 2022.

## Builds on (most cited first, 8 of 8)

- [Neural networks for post-processing ensemble weather forecasts](https://mlsanity.com/paper/1805.09091/) (2018)
- [Distributional Random Forests: Heterogeneity Adjustment and Multivariate Distributional Regression](https://mlsanity.com/paper/2005.14458/) (2020)
- [Machine learning methods for postprocessing ensemble forecasts of wind gusts: A systematic comparison](https://mlsanity.com/paper/2106.09512/) (2021)
- [Isotonic Distributional Regression](https://mlsanity.com/paper/1909.03725/) (2019)
- [Combining predictive distributions for statistical post-processing of ensemble forecasts](https://mlsanity.com/paper/1607.08096/) (2016)
- [Monotone Least Squares and Isotonic Quantiles](https://mlsanity.com/paper/1901.02398/) (2019)
- [Evaluating probabilistic forecasts of extremes using continuous ranked probability score distributions](https://mlsanity.com/paper/1905.04022/) (2019)
- [Forest-based methods and ensemble model output statistics for rainfall ensemble forecasting](https://mlsanity.com/paper/1711.10937/) (2017)

## Built on by (most cited first, 3 of 3)

- [Proper scoring rules for estimation and forecast evaluation](https://mlsanity.com/paper/2504.01781/) (2025)
- [Split Conformal Prediction in the Function Space with Neural Operators](https://mlsanity.com/paper/2509.04623/) (2025)
- [Error Analysis of Neural-Network-Based Engression](https://mlsanity.com/paper/2607.27723/) (2026)

## Abstract (arXiv)

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.