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

Alvoni, Elisabetta; Papini, Pier Luigi

Commentat. Math. Univ. Carol. 48, No. 2, 311-319 (2007).

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.

Reviewer: Jiří Anděl (Praha)

MSC:

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