JID:FSS AID:6533 /FLA [m3SC+; v 1.191; Prn:8/05/2014; 16:37] P.1(1-14) Available online at www.sciencedirect.com ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss Perturbation of bivariate copulas Radko Mesiar a , Magda Komorníková a,∗ , Jozef Komorník b a Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 813 68 Bratislava, Slovakia b Faculty of Management, Comenius University, Odbojárov 10, P.O. BOX 95, 820 05 Bratislava, Slovakia Received 3 December 2013; received in revised form 7 March 2014; accepted 21 April 2014 Abstract New types of constructions of bivariate copulas are introduced, discussed and exemplified. Based on a given copula C, we look for its perturbation into another copula C H , possibly close to C. A special stress is put on the perturbation of the basic copulas M, W , Π . Our results generalize several methods known from the literature, such as the Farlie–Gumbel–Morgenstern copula family, for example. An illustrative example when fitting copulas to real data is also added.  2014 Elsevier B.V. All rights reserved. Keywords: Farlie–Gumbel–Morgenstern copulas; Perturbation of product copula; Radially symmetric copula; Construction method for copula 1. Introduction Copulas, especially their bivariate form, have obtained a growing interest in the last two decades, especially because of their numerous applications. Recall only that a function C :[0, 1] 2 →[0, 1] is called a (bivariate) copula [24] whenever it is i) 2-increasing, i.e., V C ( [u 1 ,u 2 ]×[v 1 ,v 2 ] ) = C(u 1 ,v 1 ) + C(u 2 ,v 2 ) − C(u 1 ,v 2 ) − C(u 2 ,v 1 ) ≥ 0 for all 0 ≤ u 1 ≤ u 2 ≤ 1, 0 ≤ v 1 ≤ v 2 ≤ 1 (recall that V C ([u 1 ,u 2 ]×[v 1 ,v 2 ]) is the C -volume of the rectangle [u 1 ,u 2 ]×[v 1 ,v 2 ]); ii) grounded, i.e., C(u, 0) = C(0,v) = 0 for all u, v ∈[0, 1]; iii) it has a neutral element e = 1, i.e., C(u, 1) = u and C(1,v) = v for all u, v ∈[0, 1]. For more details, examples and applications we recommend monographs Joe (1997) [13] and Nelsen (2006) [22]. Fitting of an appropriate copula to real data is one of major tasks in application of copulas. For this purpose, a large * Corresponding author. E-mail addresses: Radko.Mesiar@stuba.sk (R. Mesiar), Magdalena.Komornikova@stuba.sk (M. Komorníková), Jozef.Komornik@fm.uniba.sk (J. Komorník). http://dx.doi.org/10.1016/j.fss.2014.04.016 0165-0114/ 2014 Elsevier B.V. All rights reserved.