Transformed Pair Copula Construction of Pareto Copula and Applications

Abstract

The study introduced the transformed copula models and a D-vine structure for three variables. The numerical applications of the models are evaluated using two different sets of real-life data that exhibit nearly no dependence, highly dependent, over, and under dispersed characteristics. We examined only the volatilities of the first data using the exponentiated weighted moving average (EWMA). The parameter estimates of the models were obtained based on the maximum likelihood estimation method for the bivariate copula models and the Dissmann algorithm for sequential top-down estimation for the D-vine structure. The results showed that the introduced copula models outperformed some existing copula models in terms of their fit statistics for both real-life and simulated data sets. In addition, the Gaussian copula model gave a better fit to the D-vine structure than some existing copula models and could be recommended for modeling a D-vine structure comprising of variables that are positively weak correlated and highly correlated.

Keywords

• Copula function
• Pair copula
• Regular vine
• Transformed copula
• Kendall's tau

References

1. [1] Sklar, M., “Fonctions de repartition an dimensions et leurs marges”, Public Institute of Statistics, Univiversity of Paris, 8: 229–231, (1959).
2. [2] Nelsen, R.B., “An Introduction to Copulas Lecture Notes in Statistics 139”, Springer Verlag New York, (1999).
3. [3] TovarCuevas, J.R., Portilla Y.J., Achcar, J.A., “A method to select bivariate copula Functions”, Revista Colombiana de Estadística, 42(1): 61-80, (2019).
4. [4] Achcar, J.A., Moala, F.A., Tarumoto, M.H., Coladello, L.F., “A bivariate generalized exponential distribution derived from copula functions in the presence of censored data and covariates”, Pesquisa Operacional, 35(1): 165-186, (2015).
5. [5] Kuvattana, S., Busababodhin, P., Areepong, Y., Sukparungsee, S., “Bivariate copulas on the exponentially weighted moving average control chart”, Songklanakarin Journal of Science & Technology, 38(5): 569-574, (2016).
6. [6] Çelebioglu, S., “Generating copulas by analytical means”, Hacettepe Bulletin of Natural Sciences and Engineering, 26: 153-173, (1997).
7. [7] Mukherjee, S., Lee, Y., Kim, J.M., Jang, J., Park, J.S., “Construction of bivariate asymmetric copulas”, Communications for Statistical Applications and Methods, 25(2): 217-234, (2018).
8. [8] Kim, D., Kim, J.M., “Analysis of directional dependence using asymmetric copula-based regression models”, Journal of Statistical Computation and Simulation, 84(9): 1990-2010, (2014).

9. [9] Latif, S., Mustafa, F., “copula-based multivariate flood probability construction: A Review”, Arabian Journal of Geosciences, 13(3): 1-25, (2020).
10. [10] Oh, R., Ahn, J.Y., Lee, W., “On copula-based collective risk models: from elliptical copulas to vine copulas”, Scandinavian Actuarial Journal, 2021(1): 1-33, (2021).
11. [11] Susam, S. O., “A new family of archimedean copula via trigonometric generator function”, Gazi University Journal of Science, 33(3): 806-813, (2020).
12. [12] Najjari, V., “The dependence structure between the monthly minimum and maximum barometric data in Iran”, Gazi University Journal of Science, 29(3): 593-597, (2016).
13. [13] Bedford, T., Cooke, R.M., “Probability density decomposition for conditionally dependent random variables modeled by vines”, Annals of Mathematics and Artificial Intelligence, 32: 245–268, (2001).
14. [14] Bedford, T., Cooke, R.M., “Vines - a new graphical model for dependent random variables", Annals of Statistics, 30 (4): 1031–1068, (2002).
15. [15] Aas, K., Czado, C., Frigessi, A., Bakken, H., “Pair-copula constructions of multiple Dependence", Insurance: Mathematics and Economics, 44(2): 182-198, (2009).
16. [16] Zhao, Y., Liu, Q., Kuang, J., Xie, K., Du, W., “Modeling multivariate dependence by nonparametric pair-copula construction in composite system reliability evaluation”, International Journal of Electrical Power & Energy Systems, 124: 106-373, (2021).
17. [17] Deng, Y., Chaganty, N.R., “Pair-copula Models for Analyzing Family Data”, Journal of Statistical Theory and Practice, 15(1): 1-12, (2021).
18. [18] Takeuchi, T.T., Kono, K.T., “Constructing a multivariate distribution function with a vine copula: towards multivariate luminosity and mass functions”, Monthly Notices of the Royal Astronomical Society, 498(3): 4365-4378, (2020).
19. [19] Valdez, E.A., Xiao, Y., “On the distortion of a copula and its margins”, Scandinavian Actuarial Journal, 2011(4): 292-317, (2011).
20. [20] Durante, F., Foschi, R., Sarkoci, P., “Distorted copulas: constructions and tail Dependence”, Communications in Statistics-Theory and Methods, 39(12): 2288-2301, (2010).
21. [21] Dißmann, J., Brechmann, E., Czado, C., Kurowicka, D., “Selecting and estimating regular vine copulae and application to financial returns”, Computational Statistics and Data Analysis, 59(1): 52-69, (2013).