Estimation methods for non-homogeneous regression models: Minimum continuous ranked probability score vs. maximum likelihood

Manuel Gebetsberger*, Jakob W. Messner, Georg J. Mayr, Achim Zeileis

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

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Non-homogeneous regression models are widely used to statistically postprocess
numerical ensemble weather prediction models. Such regression
models are capable of forecasting full probability distributions and correct
for ensemble errors in the mean and variance. To estimate the corresponding
regression coefficients, minimization of the continuous ranked probability
score (CRPS) has widely been used in meteorological post-processing studies
and has often been found to yield more calibrated forecasts compared to
maximum likelihood estimation. From a theoretical perspective, both estimators
are consistent and should lead to similar results, provided the correct
distribution assumption about empirical data. Differences between the estimated
values indicate a wrong specification of the regression model. This
study compares the two estimators for probabilistic temperature forecasting
with non-homogeneous regression, where results show discrepancies for the
classical Gaussian assumption. The heavy-tailed logistic and Student-t distributions
can improve forecast performance in terms of sharpness and calibration,
and lead to only minor differences between the estimators employed.
Finally, a simulation study confirms the importance of appropriate distribution
assumptions and shows that for a correctly specified model the maximum
likelihood estimator is slightly more efficient than the CRPS estimator.
KEYWORDS: ensemble post-processing, maximum likelihood, CRPS minimization,
probabilistic forecasting, distributional regression models
Original languageEnglish
JournalMonthly Weather Review
Issue number12
Pages (from-to)4323-4338
Number of pages16
Publication statusPublished - 2018

Bibliographical note

This article is licensed under a Creative Commons Attribution 4.0 license (


  • Ensemble post-processing
  • Maximum likelihood
  • CRPS minimization
  • Probabilistic forecasting
  • distributional regression models

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