Ordinal sums: From triangular norms to bi- and multivariate copulas

Abstract

Following a historical overview of the development of ordinal sums, a presentation of the most relevant results for ordinal sums of triangular norms and copulas is given (including gluing of copulas, orthogonal grid constructions and patchwork operators). The ordinal sums of copulas considered here are constructed not only by means of the comonotonic copula, but also by using the lower Fréchet-Hoeffding bound and the independence copula. We provide alternative proofs to some results on ordinal sums, elaborate properties common to all or just some of the ordinal sums discussed. Also included are a discussion of the relationship between ordinal sums of copulas and the Markov product and an overview of ordinal sums of multivariate copulas, illustrating aspects to be considered when extending concepts for ordinal sums of bivariate copulas to the multivariate case.