Basic bounds of Fréchet classes

Jaroslav Skřivánek

Kybernetika (2014)

  • Volume: 50, Issue: 1, page 95-108
  • ISSN: 0023-5954

Abstract

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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.

How to cite

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Skřivánek, Jaroslav. "Basic bounds of Fréchet classes." Kybernetika 50.1 (2014): 95-108. <http://eudml.org/doc/261168>.

@article{Skřivánek2014,
abstract = {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.},
author = {Skřivánek, Jaroslav},
journal = {Kybernetika},
keywords = {algebraic bound; basic bound; copula; Diophantine equation; Fréchet class; pointed convex polyhedral cone; algebraic bound; basic bound; copula; Diophantine equation; Frechet class; pointed convex polyhedral cone},
language = {eng},
number = {1},
pages = {95-108},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Basic bounds of Fréchet classes},
url = {http://eudml.org/doc/261168},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Skřivánek, Jaroslav
TI - Basic bounds of Fréchet classes
JO - Kybernetika
PY - 2014
PB - Institute of Information Theory and Automation AS CR
VL - 50
IS - 1
SP - 95
EP - 108
AB - 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.
LA - eng
KW - algebraic bound; basic bound; copula; Diophantine equation; Fréchet class; pointed convex polyhedral cone; algebraic bound; basic bound; copula; Diophantine equation; Frechet class; pointed convex polyhedral cone
UR - http://eudml.org/doc/261168
ER -

References

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  2. Embrechts, P., Lindskog, F., McNeil, A., 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. 
  3. Embrechts, P., 10.1111/j.1539-6975.2009.01310.x, J. Risk and Insurance 76 (2009), 3, 639-650. DOI10.1111/j.1539-6975.2009.01310.x
  4. Joe, H., Multivariate Models and Dependence Concepts., Chapman and Hall, London 1997. Zbl0990.62517MR1462613
  5. Nelsen, R. B., Introduction to Copulas. Second edition., Springer-Verlag, New York 2006. MR2197664
  6. Sebö, A., 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. 
  7. Skřivánek, J., Bounds of general Fréchet classes., Kybernetika 48 (2012), 1, 130-143. Zbl1251.60015MR2932932
  8. Stanley, R. P., Enumerative Combinatorics 1. Second edition., Cambridge University Press, New York 2012. MR2868112
  9. Tomás, A. P., Filgueiras, M., 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. Zbl0884.11020MR1703009

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