Abstract
In this note, we discuss a generalization of a subclass of the Rodríguez-Lallena and Úbeda-Flores family of copulas, through order statistics. This generalized family includes an important class of the Farlie-Gumbel*Morgenstern (FGM) family of copulas, which are widely used in statistics. We derive the expressions for regression function and cumulative conditional expectation function. The explicit expressions for some measures of dependence like Spearman’s rho, Gini’s gamma coefficient, and quadrant dependence are discussed. Finally, Spearman’s rho is numerically tabulated and compared for some generalized families of FGM copulas.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Amblard, C., and S. Girard. 2009. A new extension of bivariate FGM copulas. Metrika 70:1–17.
Bairamov, I., S. Kotz, and M. Bekci. 2001. New generalized Farlie–Gumbel–Morgenstern distributions and concomitants of order statistics. Journal of Applied Statistics 28:521–536.
Bairamov, I., and S. Kotz. 2002. Dependence structure and symmetry of Huang–Kotz FGM distributions and their extensions. Metrika 56:55–72.
Bayramoglu, K., and I. Bayramoglu (Bairamov). 2014. Baker–Lin–Huang type bivariate distributions based on order statistics. Communications in Statistics—Theory and Methods 43:1992–2006.
Bekrizadeh, H., G. A. Parham, and M. R. Zadkarmi. 2012. The new generalization of Farlie-Gumbel-Morgenstern copulas. Applied Mathematical Sciences 6:3527–3533.
Farlie, D. J. G. 1960. The performance of some correlation coefficients for a general bivariate distribution. Biometrika 47:307–323.
Fernández, M., and V. A. González-López. 2013. A copula model to analyze minimum admission scores. AIP Conference Proceedings 1558:1479–1482. Doi: 10.1063/1.4825799.
Genest, C., J. Nešlehová, and N. Ben Ghorbal. 2010. Spearman’s footrule and Gini’s gamma: A review with complements. Journal of Nonparametric Statistics 22:937–954.
Huang, J. S., and S. Kotz. 1999. Modifications of the Farlie-Gumbel-Morgenstern distributions. A tough hill to climb. Metrika 49:135–145.
Huang, J. S., and G. D. Lin. 2011. A note on the Sarmanov bivariate distributions. Applied Mathematics Computation 218:919–923.
Mari, D. D., and S. Kotz. 2001. Correlation and dependence. London, UK: Imperial College Press.
Nelsen, R. B. 1998. Concordance and Gini’s measure of association. Journal of Nonparametric Statistics 3, 227–238.
Nelsen, R. B. 2006. An Introduction to Copulas, 2nd ed., New York, NY: Springer.
Pathak, A. K., and P. Vellaisamy. 2014. Various measures of dependence of a new asymmetric generalized Farlie–Gumbel–Morgenstern copulas. Communications in Statistics—Theory and Methods (forthcoming).
Rodríguez-Lallena, J. A., and M. Úbeda-Flores. 2004. A new class of bivariate copulas. Statistics Probability Letters 66:315–325.
Sungur, E. A. 2005. Some observations on copula regression function. Communication in Statistics—Theory Methods 34:1967–1978.
Sklar, A. 1959. Fonctions de répartition án dimensions et leurs marges. Publications de l’ Institute Statistics de l’ University Paris 8:229–231.
Vellaisamy, P., and A. K. Pathak. 2014. Copulas and regression models. Journal of the Indian Statistics Association 52:113–134.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pathak, A.K., Vellaisamy, P. A note on generalized Farlie-Gumbel-Morgenstern copulas. J Stat Theory Pract 10, 40–58 (2016). https://doi.org/10.1080/15598608.2015.1064838
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1080/15598608.2015.1064838