## How non-symmetric can a copula be?(English)Zbl 1150.62027

Remember that a copula is a function $$C\: [0,1]^2\to [0,1]$$ such that (i) $$C(0,x)=C(x,0)=0$$, (ii) $$C(1 ,x)=C(x, 1) =x$$, (iii) $$C(x_1 ,y_1) + C(x_2,y_2) \geq C(x_1 ,y_2) + C(x_2,y_1)$$ for $$x_1\leq x_2$$, $$y_1\leq y_2$$. A quasi-copula $$Q$$ satisfies (i), (ii), it is non-decreasing in each component, and $| Q(x_1 ,y_1) - Q(x_2,y_2)| \leq | x_1-x_2| + | y_1-y_2|.$ A measure of non-symmetry of $$C$$ is $$d_C(x,y)=| C(x,y)-C(y,x)|$$ and a degree of non-symmetry of $$C$$ is $$\| d_C\| _{\infty } = \sup _{x,y} d_C(x,y)$$. Define $d^*(x,y)=\sup \{d_C(x,y) \, | \, \text{C is a copula}\}.$ The authors show that $d^*(x,y)= \min (| x-y| , x,y, 1-x, 1-y)$ and construct copulas which are maximally non-symmetric on certain subsets of $$[0,1]^2$$. It is proved that there is no copula and no quasi-copula which is maximally non-symmetric on the whole $$[0,1 ]^2$$.

### MSC:

 62H05 Characterization and structure theory for multivariate probability distributions; copulas

### Keywords:

quasi-copula; symmetry; opposite diagonal
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