\section{Conclusion}
In this work, we have studied the interconnections among several trade-off curves designed to evaluate the similarity between two probability distributions, namely precision-recall curves,
divergence frontiers, ROC and Lorenz curves.
If one connection was known to the authors of divergence frontiers, others appear to have eluded even the authors of the implied notions.
This is particularly striking for Lorenz and ROC curves which differ by mere symmetries. 
The interrelation between precision-recall and Lorenz curves is less direct, as it involves convex duality. 
That being said, it remains that the two notions are theoretically equivalent, and can be computed in practice from one another.

We hope that the exposed link will foster new research avenues for evaluation curves. 
To begin with, while the theoretical equivalence of Lorenz and PR curves has been demonstrated, the question of their empirical estimation has yet to be examined. 
For instance, investigations on potentially consistent estimators need consideration, especially in the non parametric case (although this will require refining the definitions to render the estimation from finite sample feasible). 
In particular, exposing rates of convergence of the estimators should be a worthwhile endeavor. 
Similar analysis has already been carried out for scalar metrics \cite{rubenstein2019practical,sriperumbudur2009integral}. 
We foresee that exploring links with divergences or integral probability metrics shall help in pursuing this undertaking for PR and Lorenz curves.
Similar research avenues are very plausible to deepen the link we have discussed between trade-off curves and practical bounds for domain adaptation.
{"config": "arxiv", "file": "2006.11809/9_conclusion.tex"}