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Is there a statistical distance between two 1-dim distribution F and G that $D(F,G)=\int(F(x)-G(x))^2dF(x)$?

Or to symmetrize it, take $D^s(F,G)=\int(F(x)-G(x))^2dF(x)+dG(x)$

If not, why? (What are the main disadvantages?)

I learned that $D_E(F,G)=\int(F(x)-G(x))^2dx$ is famous. But what about $D(F,G)=\int(F(x)-G(x))^2w(x)dx$? I want to check relevant materials for learning, but don't known how to start.

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I believe it's called Cramer-Von Mises distance.

https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93von_Mises_criterion

https://www.tandfonline.com/doi/abs/10.1080/10485252.2017.1285029?journalCode=gnst20

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  • $\begingroup$ Thank you very much! $\endgroup$ – Happy Superman Mar 27 at 3:52
  • $\begingroup$ And do you know if there is a distance with form like $\int (F(x)-G(x))^2w(x)dx$ or $\int (F(x)-G(x))^2w(x)dF(x)$ $\endgroup$ – Happy Superman Mar 27 at 3:52

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