Higher-order asymptotics for scoring rules

In this paper we discuss higher-order asymptotic expansions for proper scoring rules generalizing results for likelihood quantities, but meanwhile bring in the difficulty caused by the failure of the information identity. In particular, we derive higher-order approximations 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.