Baire category results for quasi–copulas

Fabrizio Durante 1 , Juan Fernández-Sánchez 2  and Wolfgang Trutschnig 3
  • 1 Faculty of Economics and Management, Free University of Bozen-Bolzano, Bolzano, Italy
  • 2 Grupo de Investigación de Análisis Matemático, Universidad de Almería, La Cañada de San Urbano, Almería, Spain
  • 3 Department for Mathematics, University of Salzburg, Salzburg, Austria

Abstract

The aim of this manuscript is to determine the relative size of several functions (copulas, quasi– copulas) that are commonly used in stochastic modeling. It is shown that the class of all quasi–copulas that are (locally) associated to a doubly stochastic signed measure is a set of first category in the class of all quasi– copulas. Moreover, it is proved that copulas are nowhere dense in the class of quasi-copulas. The results are obtained via a checkerboard approximation of quasi–copulas.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] C. Alsina, R. B. Nelsen, and B. Schweizer (1993). On the characterization of a class of binary operations on distribution functions. Statist. Probab. Lett. 17(2), 85–89.

  • [2] C. Alsina, M. J. Frank, and B. Schweizer (2003). Problems on associative functions. Aequationes Math. 66(1-2), 128–140.

  • [3] C. Alsina, M. J. Frank, and B. Schweizer (2006). Associative Functions. Triangular Norms and Copulas. World Scientific Publishing, Hackensack, NJ.

  • [4] C. Bernard, Y. Liu, N.MacGillivray, and J. Zhang (2013). Bounds on capital requirements for bivariate riskwith givenmarginals and partial information on the dependence. Depend. Model. 1, 37–53.

  • [5] I. Cuculescu and R. Theodorescu (2001). Copulas: diagonals, tracks. Rev. Roumaine Math. Pures Appl. 46(6), 731–742.

  • [6] F. Durante and C. Sempi (2015). Principles of Copula Theory. CRC/Chapman & Hall, Boca Raton, FL.

  • [7] F. Durante, J. Fernández-Sánchez, andW. Trutschnig (2015). A typical copula is singular. J.Math. Anal. Appl. 430(1), 517–527.

  • [8] F. Durante, J. Fernández-Sánchez, andW. Trutschnig (2016). Baire category results for exchangeable copulas. Fuzzy Set Syst. 284, 146–151.

  • [9] J. Fernández-Sánchez, J. A. Rodríguez-Lallena, and M. Úbeda-Flores (2011). Bivariate quasi-copulas and doubly stochastic signed measures. Fuzzy Set Syst. 168(1), 81–88.

  • [10] J. Fernández-Sánchez, and M. Úbeda-Flores (2014). A note on quasi-copulas and signed measures. Fuzzy Set Syst. 234(1), 109–112.

  • [11] J. Fernández-Sánchez and W. Trutschnig (2015). Conditioning based metrics on the space of multivariate copulas and their interrelation with uniform and levelwise convergence and Iterated Function Systems. J. Theoret. Probab. 28(4), 1311–1336.

  • [12] J. Fernández-Sánchez and W. Trutschnig (2016). Some members of the class of (quasi-) copulas with given diagonal from the Markov kernel perspective. Commun. Stat. Theor. Meth. 45(5), 1508–1526.

  • [13] G.A. Fredricks, R.B. Nelsen and J.A. Rodríguez-Lallena (2005). Copulaswith fractal supports. Insur.Math. Econ. 37(1), 42–48.

  • [14] C. Genest, J. J. Quesada-Molina, J. A. Rodríguez-Lallena, and C. Sempi (1999). A characterization of quasi-copulas. J. Multivariate Anal. 69(2), 193–205.

  • [15] M. Grabisch, J.-L. Marichal, R. Mesiar, and E. Pap (2009). Aggregation Functions. Encyclopedia of Mathematics and its Applications (No. 127). Cambridge University Press, New York.

  • [16] S. Greco, R. Mesiar, and F. Rindone (2016). Generalized bipolar product and sum. Fuzzy Optim. Decis. Mak. 15(1), 21–31.

  • [17] P. Hájek and R. Mesiar (2008). On copulas, quasicopulas and fuzzy logic. Soft Comput. 12(12), 123–1243.

  • [18] P. R. Halmos (1974). Measure Theory. Springer-Verlag, New York.

  • [19] I. Montes, E. Miranda, R. Pelessoni, and P. Vicig (2015). Sklar’s theorem in an imprecise setting. Fuzzy Set Syst. 278, 48–66.

  • [20] R. Nelsen, J. Quesada-Molina, B. Schweizer, and C. Sempi (1996). Derivability of some operations on distribution functions. In V. Beneš and J. Štepán (Eds.), Distributions with Fixed Marginals and Related Topics, pp 233–243. Inst. Math. Statist., Hayward, CA.

  • [21] R.B. Nelsen (2006). An Introduction to Copulas. Second edition. Springer-Verlag, New York.

  • [22] R. B. Nelsen, J. J. Quesada-Molina, J. A. Rodríguez-Lallena, and M. Úbeda-Flores (2008). On the construction of copulas and quasi-copulas with given diagonal sections. Insur. Math. Econ. 42(2), 473-483.

  • [23] R. B. Nelsen, J. J. Quesada-Molina, J. A. Rodríguez-Lallena, and M. Úbeda-Flores (2010). Quasi-copulas and signedmeasures. Fuzzy Set Syst. 161(17), 2328–2336.

  • [24] J. C. Oxtoby (1980). Measure and Category. A Survey of the Analogies between Topological and Measure Spaces. Second edition. Springer-Verlag, New York.

  • [25] J. A. Rodríguez-Lallena and M. Úbeda-Flores (2009). Some new characterizations and properties of quasi-copulas. Fuzzy Set Syst. 160(6), 717–725.

  • [26] W. Rudin (1986). Real and Complex Analysis. Third edition. McGraw-Hill Higher Education.

  • [27] S. Saminger-Platz and C. Sempi (2008). A primer on triangle functions. I. Aequationes Math. 76(3), 201–240.

  • [28] B. Schweizer and A. Sklar (2006). Probabilistic Metric Spaces. Dover Publications, Mineola, N.Y.

  • [29] M. Úbeda-Flores (2008). On the best-possible upper bound on sets of copulas with given diagonal sections. Soft Comput. 12(10), 1019-1025.

OPEN ACCESS

Journal + Issues

Dependence Modeling aims to provide a medium for exchanging results and ideas in the area of multivariate dependence modeling. Topics include Copula methods, environmental sciences, estimation and goodness-of-fit tests, extreme-value theory, limit laws, mass transportations, measures of association, multivariate distributions and tests, quantitative risk management, risk assessment, risk models, risk measures and stochastic orders and time series.

Search