Geometry of generators of triangular norms and copulas

Abstract

We study the relations between the shapes of strict triangular norms and their generators. The generators are not unique; to avoid this ambiguity, we introduce the class of balanced generators. We find conditions for their existence and a procedure of their computation. We formulate their influence on local properties of the corresponding triangular norms, in particular at the extremes of their domain. Many strict triangular norms are also copulas; thus, our results can also be applied to bivariate extreme value distributions described by some families of copulas.

Keywords

• multiplicative generator
• additive generator
• balanced generator
• frank copula
• ali-mikhail-haq copula
• extreme value distribution
• 03e72
• 20m14
• 54e70