## Quasi-concave copulas, asymmetry and transformations.(English)Zbl 1195.62074

Let $$C(x,y)$$, $$0\leq x, y \leq 1$$, be a copula. Then $$C$$ is quasi-concave if $$C(\lambda x+(1-\lambda )x', \lambda y + (1-\lambda )y')\geq \min \{C(x,y), C(x',y')\}$$ for all $$x,y,x',y',\lambda \in [0,1]$$. Further, $$C$$ is Schur-concave if $$C(x,y)\leq C(\lambda x+(1-\lambda )y, \lambda y+(1-\lambda )x)$$ for all $$x,y,\;\lambda \in [0,1]$$. Finally, consider copulas $$C$$ satisfying $$(*)$$ $$C((x+y)/2,(x+y)/2)\geq C(x,y)$$ for all $$x,y\in [0,1]$$.
The authors prove that a quasi-concave copula satisfying $$(*)$$ is Schur-concave. If $$C$$ is a copula, define $$\beta _C=\sup _{0\leq x,y\leq 1} \{| C(x,y)-C(y,x)| \}$$. It is known that $$\sup _C \beta _C =1/3$$. Consider the quantity $$\beta _Q=\sup \{\beta _C;\,C\text{is a quasi-concave copula}\}$$. It is proved that $$\beta (Q)=1/5$$ and indicated for which copulas this value is achieved. The authors also show that the class of quasi-concave copulas is preserved under some simple transformations.

### MSC:

 62H05 Characterization and structure theory for multivariate probability distributions; copulas 26B35 Special properties of functions of several variables, Hölder conditions, etc.
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