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In Copulas,

If $C(F(x) , G(y)) = \min (F(x) , G(y))$ then $F(x) = G(y)$.

Is it true or false?

Thanks.

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    $\begingroup$ I have no idea what Copulas or $C$ represent. But you want to prove $C(F(x),G(y))=\min(F(x),g(y))\implies F(x)=G(y)$ and it is probably easier to prove the equivalent property $F(x)\not=G(y)\implies C(F(x),G(y))\not=\min(F(x),g(y))$. $\endgroup$
    – xavierm02
    Jul 11, 2013 at 14:48

1 Answer 1

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It is true according to this paper http://www.ub.edu/stat/personal/cuadras/sort30.pdf

You may further look for "Frechet Hoeffding Bounds", if the copula attains any of these bounds, the variables are perfectly positively/negatively correlated, hence have same distribution if positive.

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