Abstract
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 \( \gamma \).
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References
Darsow, W. F., Nguyen, B., & Olsen, E. T. (1992). Copulas and Markov processes. Illinois Journal of Mathematics, 36(4), 600–642.
Durante, F., Foschi, R., & Sarkoci, P. (2010). Distorted copulas: Constructions and tail dependence. Communications in Statistics—Theory and Methods, 39(12), 2288–2301.
Durante, F., & Sempi, C. (2005). Copula and semicopula transforms. International Journal of Mathematics and Mathematical Sciences, 2005(4), 645–655.
Kachapova, F., & Kachapov, I. (2013). Monotonicity criteria. World Academy of Science, Engineering and Technology, 73, 1594–1599.
Kimeldorf, G., & Sampson, A. R. (1978). Monotone dependence. The Annals of Statistics, 6(4), 895–903.
Meyer, C. (2013). The bivariate normal copula. Communications in Statistics—Theory and Methods, 42(13), 2402–2422.
Nelsen, R. B. (2006). An introduction to copulas (2nd ed.). New York: Springer.
Schweizer, B., & Wolff, E. F. (1981). On nonparametric measures of dependence for random variables. The Annals of Statistics, 9(4), 879–885.
Stoimenov, P. A. (2008). A measure of mutual complete dependence. Ph.D. Thesis, TU Dormund.
Tran, H. D., Pham, U. H., Ly, S., & Vo-Duy, T. (2015). A new measure of monotone dependence by using Sobolev norms for copula. In Integrated uncertainty in knowledge modelling and decision making (pp. 126–137). Berlin: Springer.
Wang, S. S. (2000). A class of distortion operators for pricing financial and insurance risks. Journal of Risk and Insurance, 67(1), 15–36.
Yan, J. (2007). Enjoy the joy of copulas: With a package copula. Journal of Statistical Software, 21(4), 1–21.
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This research is funded by Vietnam National University Ho Chi Minh City (VNU-HCM) under Grant No. C2016-34-02.
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Tran, H.D., Pham, U.H., Ly, S. et al. Extraction dependence structure of distorted copulas via a measure of dependence. Ann Oper Res 256, 221–236 (2017). https://doi.org/10.1007/s10479-017-2487-2
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DOI: https://doi.org/10.1007/s10479-017-2487-2