Abstract
We propose a new family of copulas generalizing the Farlie–Gumbel–Morgenstern family and generated by two univariate functions. The main feature of this family is to permit the modeling of high positive dependence. In particular, it is established that the range of the Spearman’s Rho is [ − 3/4,1] and that the upper tail dependence coefficient can reach any value in [0,1]. Necessary and sufficient conditions are given on the generating functions in order to obtain various dependence properties. Some examples of parametric subfamilies are provided.
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References
Amblard C, Girard S (2002) Symmetry and dependence properties within a semiparametric family of bivariate copulas. Nonparametric Stat 14(6): 715–727
Amblard C, Girard S (2005) Estimation procedures for a semiparametric family of bivariate copulas. J Comput Graph Stat 14(2): 1–15
Bairamov I, Kotz S (2002) Dependence structure and symmetry of Huang-Kotz FGM distributions and their extensions. Metrika 56: 55–72
Cuadras CM, Augé J (1981) A continuous general multivariate distribution and its properties. Commun Stat Theory Methods 10: 339–353
Durante F (2006) A new class of symmetric bivariate copulas. Nonparametric Stat 18: 499–510
Embrechts P, Klüppelberg C, Mikosch T (1997) Modelling extremal events. Springer
Farlie DGJ (1960) The performance of some correlation coefficients for a general bivariate distribution. Biometrika 47: 307–323
Fischer M, Klein I (2007) Constructing Symmetric Generalized FGM Copulas by means of certain Univariate Distributions. Metrika 65: 243–260
Genest C, MacKay J (1986) Copules archimédiennes et familles de lois bidimensionnelles dont les marges sont données. Can J Stat 14: 145–159
Gumbel EJ (1960) Bivariate Exponential distributions. J Am Stat Assoc 55: 698–707
Huang JS, Kotz S (1984) Correlation structure in iterated Farlie-Gumbel-Morgenstern distributions. Biometrika 71: 633–636
Huang JS, Kotz S (1999) Modifications of the Farlie-Gumbel-Morgenstern distribution. A tough hill to climb. Metrika 49: 135–145
Joe H (1997) Multivariate models and dependence concepts. In: Monographs on statistics and applied probability, vol 73. Chapman & Hall, London
Kim J-M, Sungur EA (2004) New class of bivariate copulas. In: Proceedings for the Spring Conference 2004, Korean Statistical Society, pp 207–212
Lai CD, Xie M (2000) A new family of positive quadrant dependence bivariate distributions. Stat Probab Lett 46: 359–364
Lee MT (1996) Properties and applications of the Sarmanov family of bivariate distributions. Commun Stat Theory Methods 25(6): 1207–1222
Lehmann EL (1966) Some concepts of dependence. Ann Math Statist 37: 1137–1153
Morgenstern D (1956) Einfache Beispiele zweidimensionaler Verteilungen. Mitteilungsblatt für Mathematische Statistik 8: 234–235
Nelsen RB (1991) Copulas and association. In: Dall’Aglio G, Kotz S, Salineti G(eds) Advances and probability distribution with given marginals. Kluwer academic Publishers, Dordrecht
Nelsen RB (1993) Some concepts of bivariate symmetry. Nonparametric Stat 3: 95–101
Nelsen RB (2006) An introduction to copulas, 2nd edn. Springer Series in Statistics, Springer
Nelsen RB, Quesada-Molina JJ, Rodríguez-Lallena JA (1997) Bivariate copulas with cubic sections. Nonparametric Stat 7: 205–220
Quesada-Molina JJ, Rodríguez-Lallena JA (1995) Bivariate copulas with quadratic sections. Nonparametric Stat 5: 323–337
Rodríguez-Lallena JA (1992) Estudio de la compabilidad y diseño de nuevas familias en la teoria de cópulas. Aplicaciones. Tesis doctoral, Universidad de Granada
Rodríguez-Lallena JA, Úbeda-Flores M (2004) A new class of bivariate copulas. Stat Probab Lett 9(5): 315–325
Sarmanov OV (1966) Generalized normal correlation and two-dimensional Fréchet classes. Doklady Akademii Nauk SSSR 168(1): 596–599
Shubina M, Lee MT (2004) On maximum attainable correlation and other measures of dependence for the Sarmanov family of bivariate distributions. Commun Stat Theory Methods 33(5): 1031–1052
Schweizer B, Wolff EF (1981) On nonparametric measures of dependence for random variables. Ann Stat 9: 879–885
Sklar A (1959) Fonctions de répartition à n dimensions et leurs marges. Publ Inst Stat Univ Paris 8: 229–231
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Amblard, C., Girard, S. A new extension of bivariate FGM copulas. Metrika 70, 1–17 (2009). https://doi.org/10.1007/s00184-008-0174-7
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DOI: https://doi.org/10.1007/s00184-008-0174-7