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Dependence Modeling

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An analysis of the Rüschendorf transform - with a view towards Sklar’s Theorem

Frank Oertel
  • Deloitte LLP, Audit - Banking & Capital Markets, Hill House, 1 Little New Street, London, EC4A 3TR, UK
  • Other articles by this author:
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Published Online: 2015-09-09 | DOI: https://doi.org/10.1515/demo-2015-0008


We revisit Sklar’s Theorem and give another proof, primarily based on the use of right quantile functions. To this end we slightly generalise the distributional transform approach of Rüschendorf and facilitate some new results including a rigorous characterisation of an almost surely existing “left-invertibility” of distribution functions.

Keywords: Copulas, distributional transform; generalised inverse functions; Sklar’s Theorem

MSC:: 26A27; 60E05; 60A99; 62H05


  • [1] S. Ahmed, U. Çakmak and A. Shapiro. Coherent risk measures in inventory problems. European J. Oper. Res., 182 (1), 226-238 (2007). Web of ScienceGoogle Scholar

  • [2] R. B. Ash and C. A. Doléans-Dade. Probability and Measure Theory - 2nd Edition. Academic Press (2000). Google Scholar

  • [3] P. Billingsley. Probability and Measure - 3rd Edition. John Wiley & Sons (1995). Google Scholar

  • [4] F. Durante, J. Fernández-Sánchez and C. Sempi. A topological proof of Sklar’s theorem. Appl.Math. Lett. 26, 945-948 (2013). Web of ScienceCrossrefGoogle Scholar

  • [5] P. Embrechts and M. Hofert. A note on generalized inverses. Math. Methods Oper. Res., 77 (3), 423-432 (2013). Google Scholar

  • [6] H. Föllmer and A. Schied. Stochastic Finance: An Introduction in Discrete Time - 3rd Edition. De Gruyter Textbook (2011). Google Scholar

  • [7] M. Fréchet. Sur les tableaux de corrélation dont les marges sont donnés. Ann. Univ. Lyon, Science 4, 13-84 (1951). Google Scholar

  • [8] E. P. Klement, R. Mesiar and E. Pap. Quasi- and pseudo-inverses of monotone functions, and the construction of t-norms. Fuzzy Set. Syst., 104(1), 3-13 (1999). Google Scholar

  • [9] C. Feng, J. Kowalski, X. M. Tu and H. Wang. A Note on Generalized Inverses of Distribution Function and Quantile Transformation. Applied Mathematics, Scientific Research Publishing, 3 (12A), 2098-2100 (2012). Google Scholar

  • [10] J. F. Mai and M. Scherer. Simulating Copulas. Imperial College Press, London (2012). Google Scholar

  • [11] D. S. Moore and M. C. Spruill. Unified large-sample theory of general Chi-squared statistics for tests of fit. Ann. Statist., 3, 599-616 (1975). Google Scholar

  • [12] L. Rüschendorf. On the distributional transform, Sklar’s Theorem, and the empirical copula process. J. Statist. Plann. Inference 139(11), 3921-3927 (2009). Google Scholar

  • [13] B. Schweizer and A. Sklar. Operations on distribution functions not derivable from operations on random variables. Studia Math. 52, 43-52 (1974). Google Scholar

  • [14] B. Schweizer and A. Sklar. Probabilistic metric spaces. North-Holland, New York (1983). Google Scholar

  • [15] A. Sklar. Fonctions de répartition à n dimensions et leursmarges. Publications de l’Institut Statistique de l’Université de Paris 8, 229-231 (1959). Google Scholar

About the article

Received: 2015-03-11

Accepted: 2015-08-12

Published Online: 2015-09-09

Citation Information: Dependence Modeling, Volume 3, Issue 1, ISSN (Online) 2300-2298, DOI: https://doi.org/10.1515/demo-2015-0008.

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© 2015 Frank Oertel. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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