Kybernetika 50 no. 1, 95-108, 2014

Basic bounds of Fréchet classes

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

Abstract:

Algebraic bounds of Fr\'{e}chet 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 Fr\'{e}chet--Hoeffding bounds. We provide conditions of compatibility and propose tools for an explicit description of the basic bounds of simple Fr\'{e}chet classes.

Keywords:

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

Classification:

60E05, 62H20, 11D75

paper.pdf

References:

  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