Higher-order asymptotics for scoring rules

In this paper we discuss higher-order asymptotic expansions for proper scoring rules gen- eralizing results for likelihood quantities, but meanwhile bring in the difficulty caused by the failure of the information identity. In particular, we derive higher-order approxima- tions to the distribution of the scoring rule estimator, of the scoring rule ratio test statistic and, for a scalar parameter of interest, of the signed scoring rule root statistic. From these expansions, a modified signed scoring rule root statistic is proposed. Examples are given illustrating the accuracy of the modified signed scoring rule root statistic with respect to first-order methods.