Forecast score distributions with imperfect observations

The classical paradigm of scoring rules is to discriminate between two different forecasts by comparing them with observations. The probability distribution of the observed record is assumed to be perfect as a verification benchmark. In practice, however, observations are almost always tainted by errors and uncertainties. If the yardstick used to compare forecasts is imprecise, one can wonder whether such types of errors may or may not have a strong influence on decisions based on classical scoring rules. We propose a new scoring rule scheme in the context of models that incorporate errors of the verification data. We rely on existing scoring rules and incorporate uncertainty and error of the verification data through a hidden variable and the conditional expectation of scores when they are viewed as a random variable. The proposed scoring framework is compared to scores used in practice, and is expressed in various setups, mainly an additive Gaussian noise model and a multiplicative Gamma noise model. By considering scores as random variables one can access the entire range of their distribution. In particular we illustrate that the commonly used mean score can be a misleading representative of the distribution when this latter is highly skewed or have heavy tails. In a simulation study, through the power of a statistical test and the computation of Wasserstein distances between scores distributions, we demonstrate the ability of the newly proposed score to better discriminate between forecasts when verification data are subject to uncertainty compared with the scores used in practice. Finally, we illustrate the benefit of accounting for the uncertainty of the verification data into the scoring procedure on a dataset of surface wind speed from measurements and numerical model outputs.