Quasi-concave copulas, asymmetry and transformations

summary:In this paper we consider a class of copulas, called quasi-concave; we compare them with other classes of copulas and we study conditions implying symmetry for them. Recently, a measure of asymmetry for copulas has been introduced and the maximum degree of asymmetry for them in this sense has been computed: see Nelsen R.B., {\it Extremes of nonexchangeability\/}, Statist. Papers {\bf 48} (2007), 329--336; Klement E.P., Mesiar R., {\it How non-symmetric can a copula be\/}?, Comment. Math. Univ. Carolin. {\bf 47} (2006), 141--148. Here we compute the maximum degree of asymmetry that quasi-concave copulas can have; we prove that the supremum of $\{|C(x,y)-C(y,x)|; x,y$ in $[0,1]$; $C$ is quasi-concave\} is $\frac{1}{5}$. Also, we show by suitable examples that such supremum is a maximum and we indicate copulas for which the maximum is achieved. Moreover, we show that the class of quasi-concave copulas is preserved by simple transformations, often considered in the literature