Geodesic
Quasi-concave copulas, asymmetry and transformations
Commentationes Mathematicae Universitatis Carolinae, Tome 48 (2007) no. 2, pp. 311-319.

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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.
Classification : 26B35, 62H05
Keywords: copula; quasi-concave; asymmetry 
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Alvoni, Elisabetta; Papini, Pier Luigi. Quasi-concave copulas, asymmetry and transformations. Commentationes Mathematicae Universitatis Carolinae, Tome 48 (2007) no. 2, pp. 311-319. http://geodesic.mathdoc.fr/item/CMUC_2007__48_2_a12/