Bootstrap adjustments of signed scoring rule root statistics

Scoring rules give rise to methods for statistical inference and are useful tools to achieve robustness or reduce computations. Scoring rule inference is generally per- formed through first-order approximations to the distribution of the scoring rule es- timator or of the ratio-type statistic. In order to improve the accuracy of first-order methods even in simple models, we propose bootstrap adjustments of signed scoring rule root statistics for a scalar parameter of interest in presence of nuisance parameters. The method relies on the parametric bootstrap approach that avoids onerous calcula- tions specific of analytical adjustments. Numerical examples illustrate the accuracy of the proposed method.