Scoring rules assess the quality of probabilistic forecasts, by assigning a
numerical score based on the predictive distribution and on the event or value
that materializes. A scoring rule is proper if it encourages truthful
reporting. It is local of order k if the score depends on the predictive
density only through its value and the values of its derivatives of order up to
k at the realizing event. Complementing fundamental recent work by Parry,
Dawid and Lauritzen, we characterize the local proper scoring rules of order 2
relative to a broad class of Lebesgue densities on the real line, using a
different approach. In a data example, we use local and nonlocal proper scoring
rules to assess statistically postprocessed ensemble weather forecasts.Comment: Published in at http://dx.doi.org/10.1214/12-AOS973 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
