  We discuss weighted scoring rules for forecast evaluation and their
connection to hypothesis testing. First, a general construction principle for
strictly locally proper weighted scoring rules based on conditional densities
and scoring rules for probability forecasts is proposed. We show how
likelihood-based weighted scoring rules from the literature fit into this
framework, and also introduce a weighted version of the Hyv\"arinen score,
which is a local scoring rule in the sense that it only depends on the forecast
density and its derivatives at the observation, and does not require evaluation
of integrals. Further, we discuss the relation to hypothesis testing. Using a
weighted scoring rule introduces a censoring mechanism, in which the form of
the density is irrelevant outside the region of interest. For the resulting
testing problem with composite null - and alternative hypotheses, we construct
optimal tests, and identify the associated weighted scoring rule. As a
practical consequence, using a weighted scoring rule allows to decide in favor
of a forecast which is superior to a competing forecast on a region of
interest, even though it may be inferior outside this region. A simulation
study and an application to financial time-series data illustrate these
findings.
