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

Frank Oertel 1
  • 1 Deloitte LLP, Audit - Banking & Capital Markets, Hill House, 1 Little New Street, London, EC4A 3TR, UK


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.

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

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

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

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

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

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

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

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

  • [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).

  • [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).

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

  • [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).

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

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

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

  • [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).


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.