Concentration and ROC Curves, Revisited

This work is aimed at illustrating the strict relationship between a general definition of concentration function appeared quite some time ago on this journal and a widely used measure of the diagnostic strength of a family of binary classifiers indexed by a threshold parameter, the so-called ROC curve. The ROC curve is a common work tool in Statistics, Machine Learning and Artificial Intelligence, appearing in many applications where a binary classification (diagnosis) procedure is of interest. Hence, it is worth remarking that diagnostic strength and concentration are two sides of the same coin: the higher the concentration of one probability measure with respect to another, the higher the diagnostic strength of the likelihood ratio classification rule.