Shock models with recovery option via the maxmin copulas

Ever since the historical Sklar's theorem in 1959 copulas have been one of the main tools in modelling dependence of random variables. With the range of applications in applied mathematics expanding and varying from mathematics of finance through system theory to fuzzy set theory, there is a growing need for new types of copulas that could serve as appropriate models in these applications. It is our aim to turn a new page in constructing copulas by setting a counterpart to the famous Marshall copulas (an extension of Marshall-Olkin copulas) that are typically applied to model lifetime of a two-component system. Even a small but essential change of assumptions that the model is applied to such a system with one of the components having a recovery option turns into a substantially different problem on the level of copulas. We give a full study of the augmented case by introducing a new type of copulas, called maxmin, together with concrete applications in server-side web framework, in an economic model and in fuzzy set theory.

[1]  Fabrizio Durante,et al.  Multivariate Hierarchical Copulas with Shocks , 2010 .

[2]  Radko Mesiar,et al.  New families of symmetric/asymmetric copulas , 2014, Fuzzy Sets Syst..

[3]  I. Olkin,et al.  A Multivariate Exponential Distribution , 1967 .

[4]  José Juan Quesada-Molina,et al.  On the construction of copulas and quasi-copulas with given diagonal sections , 2008 .

[5]  Tae-Young Heo,et al.  Generalized bivariate copulas and their properties , 2011, Model. Assist. Stat. Appl..

[6]  Ludger Rüschendorf,et al.  Distributions with fixed marginals and related topics , 1999 .

[7]  Radko Mesiar,et al.  On copulas, quasicopulas and fuzzy logic , 2008, Soft Comput..

[8]  Fabrizio Durante,et al.  Copula theory and its applications : proceedings of the workshop held in Warsaw, 25-26 September 2009 , 2010 .

[9]  J. Lawless.  Statistical Models and Methods for Lifetime Data , 2002 .

[10]  Marvin Rausand,et al.  System Reliability Theory: Models, Statistical Methods, and Applications , 2003 .

[11]  Manuel Úbeda-Flores,et al.  A new class of bivariate copulas , 2004 .

[12]  B. De Baets,et al.  Transitivity frameworks for reciprocal relations: cycle-transitivity versus FG-transitivity , 2005, Fuzzy Sets Syst..

[13]  Radko Mesiar,et al.  Perturbation of bivariate copulas , 2015, Fuzzy Sets Syst..

[14]  E. Luciano,et al.  Copula Methods in Finance: Cherubini/Copula , 2004 .

[15]  S. Girard,et al.  A new symmetric extension of FGM copulas , 2007 .

[16]  Radko Mesiar,et al.  On Some Construction Methods for Bivariate Copulas , 2013, AGOP.

[17]  Xiaohu Li,et al.  Multivariate Generalized Marshall–Olkin Distributions and Copulas , 2014 .

[18]  Carles M. Cuadras,et al.  A continuous general multivariate distribution and its properties , 1981 .

[19]  Radko Mesiar,et al.  Copulas with Given Diagonal Sections: Novel Constructions and Applications , 2007, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[20]  Juan Fernández-Sánchez,et al.  A copula-based family of fuzzy implication operators , 2013, Fuzzy Sets Syst..

[21]  Bernard De Baets,et al.  Meta-theorems on inequalities for scalar fuzzy set cardinalities , 2006, Fuzzy Sets Syst..

[22]  R. Nelsen.  An Introduction to Copulas (Springer Series in Statistics) , 2006 .

[23]  Christian Genest,et al.  A Characterization of Quasi-copulas , 1999 .

[24]  Juan Fernández-Sánchez,et al.  Bivariate copulas generated by perturbations , 2013, Fuzzy Sets Syst..

[25]  Przemyslaw Grzegorzewski,et al.  Probabilistic implications , 2013, Fuzzy Sets Syst..

[26]  M. Sklar.  Fonctions de repartition a n dimensions et leurs marges , 1959 .

[27]  Fabrizio Durante,et al.  New constructions of diagonal patchwork copulas , 2009, Inf. Sci..

[28]  Bernard De Baets,et al.  Bell-type inequalities for quasi-copulas , 2004, Fuzzy Sets Syst..

[29]  P. Embrechts,et al.  Chapter 8 – Modelling Dependence with Copulas and Applications to Risk Management , 2003 .

[30]  R. Mesiar,et al.  Conjunctors and their Residual Implicators: Characterizations and Construction Methods , 2007 .

[31]  Fabrizio Durante,et al.  Copula Theory and Its Applications , 2010 .

[32]  Cécile Amblard,et al.  A new extension of bivariate FGM copulas , 2009 .

[33]  Albert W. Marshall,et al.  Copulas, marginals, and joint distributions , 1996 .

[34]  R. Nelsen.  An Introduction to Copulas , 1998 .

[35]  E. Luciano,et al.  Copula methods in finance , 2004 .

[36]  A. McNeil,et al.  Common Poisson Shock Models: Applications to Insurance and Credit Risk Modelling , 2003, ASTIN Bulletin.

[37]  Jerald F. Lawless,et al.  Statistical Models and Methods for Lifetime Data: Lawless/Statistical , 2002 .

[38]  Radko Mesiar,et al.  Semilinear copulas , 2008, Fuzzy Sets Syst..

[39]  Radko Mesiar,et al.  On special fuzzy implications , 2009, Fuzzy Sets Syst..

[40]  Claudi Alsina,et al.  On the characterization of a class of binary operations on distribution functions , 1993 .