Extraction dependence structure of distorted copulas via a measure of dependence

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, $$\lambda (C),$$ λ ( 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 $$ \lambda (C) $$ λ ( 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 $$\lambda (C)$$ λ ( 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 $$g_{ PH }$$ g P H of extreme value copulas and Wang’s transform $$ g_\gamma $$ g γ of normal and product copula are studied, and we observe their dependence behaviors changing through variability of the measure $$ \lambda (C) $$ λ ( C ) . Our results show that dependence structure of distorted copulas is subject to comonotonicity as increasing the parametric $$ \gamma $$ γ .