A Compound Positively Dependent Farlie-Gumbel-Morgenstern Bivariate Copula

Abstract

In this study, we propose a two parameter Farlie-Gumbel-Morgenstern (FGM) copula that maintains membership in the family in a way while adding the extra dependence parameter to the model by using the compound method. Also, we assess the performance of the new compound FGM copula among all the most used two-parameter families of FGM copulas. The new copula performs best for the real data having moderate dependence structure.

Keywords

• Copula
• compound disribution
• fgm copula

References

1. Bairamov, I. and Bairamov, K. (2013). From the Huang Kotz FGM distribution to Baker's bivariate distribution. Journal of Multivariate Analysis, 113, 106-115.
2. Berg, D. (2009). Copula goodness-of-fit testing: an overview and power comparison. The European Journal of Finance, 15, 675-701.
3. Cossette, H., Cote, M.P., Marceau, E. and Moutanabbir, K. (2013). Multivariate distribution defined with Farlie-Gumbel-Morgenstern copula and mixed Erlang marginals: Aggregation and capital allocation. Insurance: Mathematics and Economics, 52(3), 560-572.
4. Farlie, D.J.G. (1960). The performance of some correlation coefficients for a general bivariate distribution. Biometrika, 47 (3-4), 307-23.
5. Gumbel, E. (1960). Bivariate exponential distributions. Journal of the American Statistical Association, 55(292), 698-707.
6. Huang, J.S. and Kotz, S. (1984). Correlation structure in iterated Farlie-Gumbel-Morgenstern distributions. Biometrika, 71(3), 633-636.
7. Huang, J.S. and Kotz, S. (1999). Modifications of the Farlie-Gumbel-Morgenstern distributions. A tough hill to climb. Metrika, 49(2),135-45.
8. Kelner, M., Landsman, Z. and Makov, U.E. (2021). Compound Archimedean copulas. International Journal of Statistics and Probability, 10(3), 126-126.

9. Lai, C.D. and Xie, M. (2000). A new family of positive quadrant dependent bivariate distributions. Statistics & Probability Letters, 46, 359-634.
10. Lu, L. and Ghosh, S.K. (2021). Nonparametric estimation and testing for positive quadrant dependent bivariate copula. Journal of Business and Economic Statistics, In Press.
11. Morgenstern, D. (1956). Einfache beispiele zweidimensionaler verteilungen. Mitteilingsblatt Feur Mathematische Statistik, 8, 234-235.
12. Navarro, J. and Durante, F. (2017). Copula-based representations for the reliability of the residual lifetimes of coherent systems with dependent components. Journal of Multivariate Analysis, 158, 87-102.
13. Navarro, J., Ruiz, J.M. and Sandoval, C.J. (2007). Properties of coherent systems with dependent components. Communications in Statistics-Theory and Methods, 36, 175-191.
14. Patton, A. J. (2006). Estimation of multivariate models for time series of possibly different lengths. Journal of Applied Econometrics, 21(2), 147-173.
15. Sklar, A. (1959). Fonctions de répartition à n dimensions et leurs marges. Publications de l'Institut Statistique de l'Université de Paris, 8, 229-231.