Receiver operating characteristic (ROC) curves: equivalences, beta model, and minimum distance estimation

Receiver operating characteristic (ROC) curves are used ubiquitously to evaluate scores, features, covariates or markers as potential predictors in binary problems. We characterize ROC curves from a probabilistic perspective and establish an equivalence between ROC curves and cumulative distribution functions (CDFs). These results support a subtle shift of paradigms in the statistical modelling of ROC curves, which we view as curve fitting. We propose the flexible two-parameter beta family for fitting CDFs to empirical ROC curves and derive the large sample distribution of minimum distance estimators in general parametric settings. In a range of empirical examples the beta family fits better than the classical binormal model, particularly under the vital constraint of the fitted curve being concave.