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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?

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  • $\begingroup$ 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). $\endgroup$
    – Glen_b
    Commented Jul 27, 2021 at 23:30

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See Dana Quade's pair chart which relates to the Wilcoxon-Mann-Whitney two-sample rank-sum test.

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