On the Classes of Copulas and Quasi-Copulas With a Given Diagonal Section

Abstract

Let C δ and Q δ be, respectively, the classes of all copulas and quasi-copulas whose diagonal section is δ. We determine under which conditions on δ we have: (a) both C δ and Q δ consist of a singleton; (b) C δ = Q δ . Moreover, a simple construction of copulas with a given convex diagonal section is introduced.

Maximal directions of monotonicity of an aggregation function

“…Detecting the maximal directions of monotonicity naturally leads to the transformation of a given aggregation function into the largest aggregation function that has the same maximal directions of monotonicity. In doing so, we find connections with existing construction methods for aggregation functions, and more specifically quasi-copulas and copulas with given diagonal section, a topic on which a great deal of research has already been done [9,12,15,23,31].…”

Connecting copula properties with reliability properties of coherent systems

“…Notice that, up to the case of comonotonic dependence structure, several copulas may have the same diagonal section (eg, Durante and Fernández‐Sánchez 23 and Durante and Sempi 14 ). Therefore, some special comparisons of system lifetimes with identically distributed components do not depend on the whole copula information, but just on the local properties related to the copula diagonal section.…”