Gini Score under Ties and Case Weights

The Gini score is a popular statistical tool in model validation. The Gini score has originally been introduced and used for binary responses Y {0, 1}, and there are many equivalent formulations of the (binary) Gini score such as the receiver operating curve (ROC) and the area under the curve (AUC); see, e.g., [Bamber (1975)], [Hanley-McNeil (1982)] and [Fawcett (2006)]. These different formulations are also equivalent to the Wilcoxon-Mann-Whitney's U statistic, see [Hanley-McNeil (1982)], [DeLong et al. (1988)], [Byrne (2016)], and to [Somers (1962)]'s D, see [Newson (2002)]. Thus, there are at least five equivalent formulations of the Gini score in a binary context, and there is a broad literature on its behavior which is well understood. When it comes to general real-valued responses, things become more difficult, and definitions and results on the Gini score are mainly found in the credit risk and actuarial literature. In this stream of literature, the Gini score has been introduced by [Gourieroux-Jasiak (2007)], [Frees et al. (2011), Frees et al. (2013)]. Furthermore, in the real-valued setting the Gini score is studied in much detail in [Denuit et al. (2019)] and [Denuit-Trufin (2021)]. The Gini score is a statistic that assesses whether a given risk ranking is correct.