We consider a new generalization of the skew-normal distribution introduced by Azzalini ( 1985 Azzalini , A. ( 1985 ). A class of distributions which includes the normal ones . Scand. J. Statis. 12 ( 2 ): 171 – 178 . [Web of Science ®] [Google Scholar] ). We denote this distribution Beta skew-normal (BSN) since it is a special case of the Beta generated distribution (Jones, 2004 Jones , M. C. ( 2004 ). Families of distributions of order statistics . Test 13 ( 1 ): 1 – 43 . [Crossref], [Web of Science ®] [Google Scholar] ). Some properties of the BSN are studied. We pay attention to some generalizations of the skew-normal distribution (Bahrami et al., 2009 Bahrami , W. , Agahi , H. , Rangin , H. ( 2009 ). A two-parameter Balakrishnan skew-normal distribution . J. Statist. Res. Iran 6 : 231 – 242 . [Google Scholar] ; Sharafi and Behboodian, 2008 Sharafi , M. , Behboodian , J. ( 2008 ). The Balakrishnan skew-normal density . Statist. Pap. 49 : 769 – 778 . [Crossref], [Web of Science ®] [Google Scholar] ; Yadegari et al., 2008 Yadegari , I. , Gerami , A. , Khaledi , M. J. ( 2008 ). A generalization of the Balakrishnan skew-normal distribution . Statist. Probab. Lett. 78 : 1165 – 1167 . [Crossref], [Web of Science ®] [Google Scholar] ) and to their relations with the BSN.

Scoring rules give rise to methods for statistical inference and are useful tools to achieve robustness or reduce computations. Scoring rule inference is generally performed through first-order approximations to the distribution of the scoring rule estimator 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 calculations specific of analytical adjustments. Numerical examples illustrate the accuracy of the proposed method.

Ensembles of meteorological quantities obtained from numerical models can be used for forecasting weather variables. Unfortunately, such ensembles are often biased and under-dispersed and therefore need to be post-processed. Ensemble model output statistics (EMOS) is a widely used post-processing technique to reduce bias and dispersion errors of numerical ensembles. In the EMOS approach, a full probabilistic prediction is given in the form of a predictive distribution with parameters depending on the ensemble forecast members. Parameters are then estimated and substituted, thus obtaining a so-called estimative predictive distribution. Nonetheless, estimative distributions may perform poorly in terms of the coverage probability of the corresponding quantiles. This work proposes the use of predictive distributions based on a bootstrap adjustment of estimative predictive distributions, in the context of EMOS models. These distributions are calibrated, which means that the corresponding quantiles provide exact coverage probabilities, in contrast to the estimative distributions. The introduction of the bootstrap calibrated procedure for EMOS is the innovative aspect of this study. The performance of the suggested calibrated EMOS is evaluated in two simulation studies, comparing the different predictive distributions by means of the log-score, the continuous ranked probability score, and the coverage of the corresponding predictive quantiles. The results of these simulation studies show that the proposed calibrated predictive distributions improve estimative solutions, both reducing the mean scores and producing quantiles with exact coverage levels. The good performance of the new calibrated EMOS is further stressed in two real data applications, one about maximum daily temperatures at sites located in the Veneto region (Italy) and the other one about wind speed forecasts at weather stations over Germany.