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 strict propriety. This is accomplished by applying the original scoring rule to a censored distribution, acknowledging that censoring emerges as a natural localization device due to its ability to retain precisely all relevant information of the original 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 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 superior power properties of censoring.
