August 24, 2010 16:58 WSPC/INSTRUCTION FILE Durante˙FernandezSanchez˙Revision International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems c  World Scientific Publishing Company ON THE CLASSES OF COPULAS AND QUASI-COPULAS WITH A GIVEN DIAGONAL SECTION FABRIZIO DURANTE Department of Knowledge-Based Mathematical Systems Johannes Kepler University, Linz, Austria School of Economics and Management Free University of Bozen-Bolzano, Bolzano, Italy e-mail: fabrizio.durante@unibz.it JUAN FERN ´ ANDEZ-S ´ ANCHEZ Grupo de Investigaci´ on de An´ alisisMatem´atico Universidad de Almer´ ıa, La Ca˜ nada de San Urbano, Almer´ ıa, Spain e-mail: juanfernandez@ual.es Received 12 February 2010 Revised August 24, 2010 Let C δ and Q δ be, respectively, the classes of all copulas and quasi–copulas whose diag- onal section is δ. We determine under which conditions on δ we have: (a) both C δ and Q δ consist of a singleton; (b) C δ = Q δ . Moreover, a simple construction of copulas with a given convex diagonal section is introduced. Keywords : Copula; Section of a copula; Tail dependence. 1. Introduction A two–dimensional copula 1,2 (a copula, for short) is a function C : I 2 → I (I = [0, 1]) that satisfies the following properties: (C1) C (x, 1) = C (1,x)= x for all x ∈ I; (C2) C is increasing in each variable; (C3) for all x 1 , x 2 , y 1 , y 2 in I with x 1 ≤ x 2 and y 1 ≤ y 2 , V C ([x 1 ,x 2 ] × [y 1 ,y 2 ]) = C (x 2 ,y 2 ) − C (x 1 ,y 2 ) − C (x 2 ,y 1 )+ C (x 1 ,y 1 ) ≥ 0. Condition (C3) is called the 2–increasing property of C , and V C ([x 1 ,x 2 ] × [y 1 ,y 2 ]) is called the C –volume of the rectangle [x 1 ,x 2 ] × [y 1 ,y 2 ]. According to Sklar’s Theorem, copulas are functions that express the rank- invariant dependence between a pair of continuous random variables and, as such, they have been largely used in statistical applications 1,3 . In particular, exam- ples of copulas are given by M (x, y) = min{x, y}, Π(x, y)= xy and W (x, y)= 1