Abstract
In recent years, there has been a raise of interest in the determination of copulas with given values at some fixed points, or with given horizontal, vertical, affine, diagonal, or sub-diagonal sections and combinations thereof. Closely related to these investigations are the determination and characterization of increasing and 2-increasing functions with given margins whose domain is a subset of the unit square as well as necessary and sufficient conditions providing that the combination (patchwork) of such functions on sub-domains yields a (new) 2-increasing aggregation function on [0,1]2, in particular a copula. In the present contribution we provide a full characterization of increasing, 2-increasing functions with prescribed margins acting on a sub-rectangle of the unit square. The characterization allows to determine easily the greatest and smallest such functions and to look at the results on copulas with given horizontal and/or vertical sections and its boundaries from a more general and unified viewpoint. We further discuss necessary and sufficient conditions for a patchwork based on triangular sub-domains.
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Durante, F., Saminger-Platz, S., Sarkoci, P. (2008). On Patchwork Techniques for 2-Increasing Aggregation Functions and Copulas. In: Dubois, D., Lubiano, M.A., Prade, H., Gil, M.Á., Grzegorzewski, P., Hryniewicz, O. (eds) Soft Methods for Handling Variability and Imprecision. Advances in Soft Computing, vol 48. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85027-4_42
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DOI: https://doi.org/10.1007/978-3-540-85027-4_42
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