Partial AUC and optimal thresholds

Extensive literature exists on how to estimate optimal thresholds based on various accuracy measures using receiver operating characteristic (ROC) and cumulative accuracy profile (CAP) curves. This paper now proposes an alternative measure to represented the specific partial area under the ROC and CAP curves. The relationship between ROC and CAP functions is examined using differential equations of the new defined partial area under curves. In addition, the relationship with the optimal thresholds under conditions of various accuracy measures for the ROC and CAP functions is also derived. We assume there are two kinds of distribution functions composing the mixed distribution as various normal distributions before finding the optimal thresholds. Corresponding type 1 and 2 errors are also explored and discussed under various conditions for accuracy measures.