Localizing Strictly Proper Scoring Rules*

When comparing predictive distributions, forecasters are typically not equally interested in all regions of the outcome space. To address the demand for focused forecast evaluation, we propose a procedure to transform strictly proper scoring rules into their localized counterparts while preserving the score divergence and strict propriety. This is accomplished by applying the original scoring rule to a censored distribution. Our procedure nests the censored likelihood score as a special case. Among a multitude of others, it also implies a class of censored kernel scores that offers a (possibly multivariate) alternative to the threshold weighted Continuously Ranked Probability Score (twCRPS), extending its local propriety to more general weight functions than single tail indicators. Within this localized framework, we obtain a generalization of the Neyman Pearson lemma, establishing the censored likelihood ratio test as uniformly most powerful. For other tests of localized equal predictive performance, results of Monte Carlo simulations and empirical applications to risk management, inflation and climate data consistently emphasize the excellent power properties of censoring versus other localization methods.