Distorted distributions and ROC curves

We consider new distortion functions based on the receiver operating characteristic (ROC) and ordinal dominance (OD) curves. Various stochastic orders and aging properties are characterized by using the interpretation of ROC and OD distortions as cumulative distribution functions of suitable relative random variables. Connections with Lorenz curve and Gini's index are examined too. We also define further distortion functions based on the concept of partial area under the ROC curve and partial area under the OD curve, by pointing out their connections with recent univariate skewed models and with the equilibrium distribution. Our theoretical findings are illustrated in different settings through real data applications of the distortion‐based approach by using a semiparametric estimation of the ROC curve.