Copula constructions using ultramodularity

Erich Peter Klement (), Anna Kolesárová (), Radko Mesiar () and Susanne Saminger-Platz ()
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Erich Peter Klement: Johannes Kepler University, Department of Knowledge-Based Mathematical Systems
Anna Kolesárová: Slovak University of Technology, Department of Information Engineering, Automation and Mathematics, Faculty of Chemical and Food Technology
Radko Mesiar: Slovak University of Technology, Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering
Susanne Saminger-Platz: Johannes Kepler University, Department of Knowledge-Based Mathematical Systems

Chapter Chapter 9 in Copulas and Dependence Models with Applications, 2017, pp 135-156 from Springer

Abstract: Abstract We discuss some copula constructions by means of ultramodular bivariate copulas. In general, the ultramodularity of a real function is a stronger version of both its convexity and its supermodularity (the latter property being always satisfied in the case of a bivariate copula). In a statistical sense, ultramodular bivariate copulas are related to random vectors whose components are mutually stochastically decreasing with respect to each other. Analytically speaking, an ultramodular bivariate copula is characterized by the convexity of all of its horizontal and vertical sections. Among other results, we give a sufficient condition for the additive generators of Archimedean ultramodular bivariate copulas, and we propose two constructions for bivariate copulas: the first one being based on ultramodular aggregation functions, and the other one showing the special role of ultramodularity and Schur concavity for a product-like composition of bivariate copulas being again a bivariate copula.

Date: 2017
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DOI: 10.1007/978-3-319-64221-5_9

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