Quadratic constructions of copulas Anna Kolesárová a , Gaspar Mayor b,⇑ , Radko Mesiar c,d a Institute of Information Engineering, Automation and Mathematics, Faculty of Chemical and Food Technology, Slovak University of Technology in Bratislava, Radlinského 9, 812 37 Bratislava 1, Slovakia b Department of Mathematics and Computer Science, University of the Balearic Islands, 07122 Palma (Mallorca), Spain c Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology in Bratislava, Radlinského 11, 813 68 Bratislava 1, Slovakia d Centre of Excellence IT4Innovations, Division University of Ostrava, IRAFM, Czech Republic article info Article history: Received 2 April 2013 Received in revised form 19 February 2015 Accepted 10 March 2015 Available online 19 March 2015 Keywords: Copula Invariant copula Plackett copula Quadratic construction abstract In the paper we introduce and study quadratic constructions of copulas based on com- position of a copula and some quadratic polynomial. We characterize all quadratic polynomials whose composition with an arbitrary copula always results in a copula. Due to this result, we can assign a two-parametric class K c;d C   of copulas with parameters ðc; dÞ in a certain subset X # R 2 to each copula C. Moreover, we also determine all copulas invariant with respect to the quadratic constructions presented. This investigation brings two interesting parametric classes of copulas. We show that the union of these two classes is equal to the so-called Plackett family of copulas. We add some properties of these copu- las and their statistical consequences. Ó 2015 Elsevier Inc. All rights reserved. 1. Introduction Copulas [23,19,9] play an important role in all areas dealing with random vectors. Fitting an appropriate copula to given experimental data requires a large buffer of copulas, and therefore any new construction method of copulas extends the possibility of their applications. There are several construction methods determining new copulas by means of two or more given copulas. For example, we recall the standard convex combinations of copulas, several types of ordinal sums, or other patchwork techniques; see e.g., [1,2,4,5,7,8,6,10,14–17,19,20,22]. A new copula can also be obtained by a transformation of the given copula C [18], or by its flipping, or by related techniques based on measure preserving transformations [9,19,12]. In all these cases, the knowledge of the values x; y and Cðx; yÞ is not sufficient for determining the value Dðx; yÞ of a new copula D constructed by means of the copula C. One of a few known methods which determine the values Dðx; yÞ directly by means of x; y and Cðx; yÞ, is the method recently introduced in [3], and also discussed in [13]. By the result proved there, for any copula C : ½0; 1 2 !½0; 1, the function D C defined on ½0; 1 2 by D C ðx; yÞ¼ Cðx; yÞðx þ y  Cðx; yÞÞ ð1Þ is a copula. Note, that if we consider a quadratic polynomial P of three variables, Pðx; y; zÞ¼ xz þ yz  z 2 , then the construc- tion (1) can be seen as the composite function D C ðx; yÞ¼ Pðx; y; Cðx; yÞÞ: ð2Þ http://dx.doi.org/10.1016/j.ins.2015.03.016 0020-0255/Ó 2015 Elsevier Inc. All rights reserved. ⇑ Corresponding author. E-mail addresses: anna.kolesarova@stuba.sk (A. Kolesárová), gmayor@uib.es (G. Mayor), radko.mesiar@stuba.sk (R. Mesiar). Information Sciences 310 (2015) 69–76 Contents lists available at ScienceDirect Information Sciences journal homepage: www.elsevier.com/locate/ins