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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.

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