Lognormal Lorenz and normal receiver operating characteristic curves as mirror images

The Lorenz curve for assessing economic inequality depicts the relation between two cumulative distribution functions (CDFs), one for the distribution of incomes or wealth and the other for their first-moment distribution. By contrast, the receiver operating characteristic (ROC) curve for evaluating diagnostic systems depicts the relation between the complements of two CDFs, one for the distribution noise and the other for the distribution of signal plus noise. We demonstrate that the lognormal model of the Lorenz curve, which is often adopted to model the distribution of income and wealth, is a mirror image of the equal-variance normal model of the ROC curve, which is a fundamental model for evaluating diagnostic systems. The relationship between these two models extends the potential application of each. For example, the lognormal Lorenz curve can be used to evaluate diagnostic systems derived from equal-variance normal distributions.