On the degree of asymmetry of a quasi-copula with respect to a curve

The geometrical interpretation of symmetry of a function on the unit square is that it takes the same value in mirror images w.r.t. the main diagonal. Here, this concept is generalized by considering the reflection w.r.t. a curve representing the graph of an automorphism of the unit interval. A function on the unit square that takes the same value in mirror images w.r.t. such a curve is called symmetric w.r.t. that curve. Moreover, a measure is proposed for quantifying to what extent a (quasi-)copula can be regarded as being asymmetric w.r.t. a given curve. The major part of the paper is concerned with establishing lower and upper bounds on this degree of asymmetry. Finally, it is shown that these bounds are sharp within the class of copulas. (C) 2018 Published by Elsevier B.V.