Can I use the ROC curve to compare two distributions?

The ROC curve for two distributions $$F$$ and $$G$$ can be defined as

$$\mbox{ROC}(u) = {F}(G^{-1}(u)),$$ for $$u \in (0,1)$$. So, if $$F=G$$, then $$\mbox{ROC}(u) = u$$. Can I use this property to compare the two distributions $$F$$ and $$G$$? For example, by checking deviations from the identity function?

• Can you explain more about how you're obtaining/observing F and/or G? Are you comparing two ecdfs or one ecdf and one theoretical cdf? Certainly if F is an ecdf and G a continuous theoretical cdf, any number of measures of deviation from a straight line would be usable as goodness of fit tests (and at least some of them would correspond to well known tests). Jul 27 at 23:30