JID:FSS AID:7848 /FLA [m3SC+; v1.328; Prn:29/04/2020; 13:50] P.1(1-10) Available online at www.sciencedirect.com ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss A comprehensive family of copulas to model bivariate random noise and perturbation Ayyub Sheikhi a,∗ , Vahid Amirzadeh a , Radko Mesiar b,c a Department of statistics, Faculty of mathematics and computer, Shahid Bahonar University, Kerman, Iran b Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, STU Bratislava, Slovakia c Palacky University Olomouc, Faculty of Science, Department of Algebra and Geometry, 17, Listopadu 12, 771 46 Olomouc, Czech Republic Received 2 September 2019; received in revised form 10 April 2020; accepted 11 April 2020 Abstract Based on the random vector (X + Z, Y + Z) we study the perturbation C X+Z,Y +Z of the copula C X,Y of the random vector (X, Y) when the random noise Z is independent of both X and Y . We also propose a bivariate copula-based modification of the Irwin-Hall distribution and use it to extend the results of Mesiar et al. (2019) and present several examples for illustration.  2020 Elsevier B.V. All rights reserved. Keywords: Copula; Noise; Perturbation of copula; Irwin-Hall distribution; Random vector 1. Introduction and preliminaries Copulas are popular and useful tools for modeling the dependence between variables. They allow one to eas- ily model and estimate the distribution of random vectors by estimating marginals and copulas separately. Briefly, stochastic dependence of a couple of random variables is described by means of copulas. Thorough and perfect intro- ductions to copulas are given by Joe (1997) [8], Durante and Sempi (2016) [6] and Nelsen (2006) [9]. By the Sklar theorem (see e.g., [9]), for any continuous random vector (X, Y) with the joint distribution function F X,Y and the marginal distribution functions F X , F Y , there exists a unique copula C : [0, 1] 2 → [0, 1] ,F X,Y (x,y) = C(F X (x),F Y (y)), x,y ∈ R. (1) We denote the copula coupling X and Y as C X,Y (u, v). (Note also that we have an axiomatic characterization of copulas, see [6,8,9].) We then have, supposing the continuity of F X and F Y , C X,Y (u,v) = F X,Y (F −1 X (u), F −1 Y (v)), (2) where F −1 X , F −1 Y :]0, 1[→ R are the related quantile functions (for more details see [6,8,9]). * Corresponding author. E-mail address: sheikhy.a@uk.ac.ir (A. Sheikhi). https://doi.org/10.1016/j.fss.2020.04.010 0165-0114/ 2020 Elsevier B.V. All rights reserved.