Annals of Operations Research

, Volume 256, Issue 2, pp 221–236 | Cite as

Extraction dependence structure of distorted copulas via a measure of dependence

  • Hien Duy Tran
  • Uyen Hoang Pham
  • Sel Ly
  • T. Vo-Duy


Copulas are one of the most powerful tools in modeling dependence structure of multivariate variables. In Tran et al. (Integrated uncertainty in knowledge modelling and decision making. Springer, Berlin, pp 126–137, 2015), we have constructed a new measure of dependence, \(\lambda (C),\) based on Sobolev norm for copula C which can be used to characterize comonotonicity, countermonotonicity and independence of random vectors. This paper aims to use the measure \( \lambda (C) \) to study how dependence structure of a distorted copula after being transformed by a distortion function is changed. Firstly, we propose two methods to estimate the measure \(\lambda (C)\), one for known copula C using conditional copula-based Monte Carlo simulation and the latter for unknown copula dealing with empirical data. Thereafter, PH-transform \(g_{ PH }\) of extreme value copulas and Wang’s transform \( g_\gamma \) of normal and product copula are studied, and we observe their dependence behaviors changing through variability of the measure \( \lambda (C) \). Our results show that dependence structure of distorted copulas is subject to comonotonicity as increasing the parametric \( \gamma \).


Copulas Distortion functions Monotone dependence Comonotonicity Countermonotonicity Measures of dependence Sobolev norm 



This research is funded by Vietnam National University Ho Chi Minh City (VNU-HCM) under Grant No. C2016-34-02.


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Hien Duy Tran
    • 1
  • Uyen Hoang Pham
    • 2
  • Sel Ly
    • 3
  • T. Vo-Duy
    • 4
    • 5
  1. 1.Tan Tao UniversityLong AnVietnam
  2. 2.University of Economics and LawHo Chi Minh CityVietnam
  3. 3.Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnam
  4. 4.Division of Computational Mathematics and Engineering, Institute for Computational ScienceTon Duc Thang UniversityHo Chi Minh CityVietnam
  5. 5.Faculty of Civil EngineeringTon Duc Thang UniversityHo Chi Minh CityVietnam

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