# Article

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Keywords:
copula; quasi-concave; asymmetry
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.
References:
[1] Durante F.: Solution of an open problem for associative copulas. Fuzzy Sets and Systems 152 (2005), 411-415. MR 2138520 | Zbl 1065.03035
[2] Genest C., Ghoudi K., Rivest L.-P.: Discussion on Understanding relationships using copulas'' by E. Frees and E. Valdez. N. Am. Actuar. J. 2 (1999), 143-149. MR 2011244
[3] Klement E.P., Mesiar R.: How non-symmetric can a copula be?. Comment. Math. Univ. Carolin. 47 (2006), 141-148. MR 2223973 | Zbl 1150.62027
[4] Klement E.P., Mesiar R., Pap E.: Different types of continuity of triangular norms revisited. New Math. Nat. Comput. 1 (2005), 1-17. MR 2158962 | Zbl 1081.26024
[5] Nelsen R.B.: An Introduction to Copulas. 2nd edition, Springer, New York, 2006. MR 2197664 | Zbl 1152.62030
[6] Nelsen R.B.: Extremes of nonexchangeability. Statist. Papers 48 (2007), 329-336. MR 2295821 | Zbl 1110.62071
[7] Robert A.W., Varberg D.E.: Convex Functions. Academic Press, New York, 1973. MR 0442824

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