Kybernetika 50 no. 1, 95-108, 2014

Basic bounds of Fréchet classes

Jaroslav SkřivánekDOI: 10.14736/kyb-2014-1-0095


Algebraic bounds of Fréchet classes of copulas can be derived from the fundamental attributes of the associated copulas. A minimal system of algebraic bounds and related basic bounds can be defined using properties of pointed convex polyhedral cones and their relationship with non-negative solutions of systems of linear homogeneous Diophantine equations, largely studied in Combinatorics. The basic bounds are an algebraic improving of the Fréchet-Hoeffding bounds. We provide conditions of compatibility and propose tools for an explicit description of the basic bounds of simple Fréchet classes.


copula, Fréchet class, Diophantine equation, algebraic bound, basic bound, pointed convex polyhedral cone


60E05, 62H20, 11D75


  1. E. Contejean and H. Devie: An efficient incremental algorithm for solving system of linear Diophantine equations. Inform. and Comput. 113 (1994), 1, 143-172.   CrossRef
  2. P. Embrechts, F. Lindskog and A. McNeil: Modelling dependence with copulas and applications to risk management. In: Handbook of Heavy Tailed Distributions in Finance (S. T. Rachev, ed.), Elsevier/North-Holland 2003.   CrossRef
  3. P. Embrechts: Copulas: A personal view. J. Risk and Insurance 76 (2009), 3, 639-650.   CrossRef
  4. H. Joe: Multivariate Models and Dependence Concepts. Chapman and Hall, London 1997.   CrossRef
  5. R. B. Nelsen: Introduction to Copulas. Second edition. Springer-Verlag, New York 2006.   CrossRef
  6. A. Sebö: Hilbert bases, Carathéodory's theorem and combinatorial optimization. In: Integer Programming and Combinatorial Optimization (R. Kannan and W. Pulleyblanck, eds.), University of Waterloo Press, Waterloo 1990, pp. 431-456.   CrossRef
  7. J. Skřivánek: Bounds of general Fréchet classes. Kybernetika 48 (2012), 1, 130-143.   CrossRef
  8. R. P. Stanley: Enumerative Combinatorics 1. Second edition. Cambridge University Press, New York 2012.   CrossRef
  9. A. P. Tomás and M. Filgueiras: An algorithm for solving systems of linear Diophantine equations in naturals. In: Progress in Artificial Intelligence (Coimbra) (E. Costa and A. Cardoso, eds.), Lecture Notes in Comput. Sci. 1323, Springer-Verlag, Berlin 1997, pp. 73-84.   CrossRef