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

Klement, Erich Peter; Mesiar, Radko

Commentat. Math. Univ. Carol. 47, No. 1, 141-148 (2006).

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 \([0,1 ]^2\).

Reviewer: Jiří Anděl (Praha)

MSC:

62H05

Characterization and structure theory for multivariate probability distributions; copulas

Keywords:

quasi-copula; symmetry; opposite diagonal