Enrique de Amo

Absolutely continuous copulas with given sub-diagonal section

Abstract Recently, Durante and Jaworski (2008) [6] have proved that the class of absolutely continuous copulas with a given diagonal section is non-empty in case that the diagonal function is such that the set of points where this coincides with the identity function has null-measure. In this paper, we show that if we consider sub-diagonals (or super-diagonals), then the framework changes. Concretely, for each sub-diagonal function there exists an absolutely continuous copula having this function as sub-diagonal section.

Characterization of copulas with given diagonal and opposite diagonal sections

In recent years special attention has been devoted to the problem of finding a copula, the diagonal section and opposite diagonal section of which are known. For given diagonal function and opposite diagonal functions, we provide necessary and sufficient conditions for the existence of a copula to have these functions as diagonal and opposite diagonal sections. We make use of techniques related to interpolation between the diagonals, the construction of checkerboard copulas, and linear programming. This result allows us to solve two open problems: to characterize the class of copulas where the knowledge of diagonal and opposite diagonal sections determines the copula in a unique way, and to formulate necessary and sufficient conditions for each pair of such functions to be the diagonal and opposite diagonal sections of a unique copula.

Moments and associated measures of copulas with fractal support

Copulas are closely related to the study of distributions and the dependence between random variables. In this paper we develop a recurrence formula for the moments of a measure associated with a copula (a bivariate distribution function with uniform one-dimensional marginals) in the case that its support is a fractal set. We do the same for its principal and secondary diagonals. We also study certain measures of dependence or association for these copulas with fractal supports.

A family of singular functions and its relation to harmonic fractal analysis and fuzzy logic

Abstract We study a parameterized family of singular functions which appears in a paper by H. Okamoto and M. Wunsch (2007). Various properties are revisited from the viewpoint of fractal geometry and probabilistic techniques. Hausdorff dimensions are calculated for several sets related to these functions, and new properties close to fractal analysis and strong negations are explored.

Constructions of copulas under prescribed sections

The main problems related to copula sections are reviewed and various methods for constructing copulas that preserve some partial information, are presented. Here, the scope of interest extends from seminal work on the existence and construction of copulas with given diagonal sections to the most recent research on copulas with given diagonal and opposite diagonal sections. Also a survey is given on the state of the art on related domains such as the construction of special families of copulas and generalizations of the copula concept.

Copulas and Self-affine Functions

We characterize self-affine functions whose graphs are the support of a copula using the fact that the functions defined on the unit interval whose graphs support a copula are those that are Lebesgue-measure-preserving. This result allows the computation of the Hausdorff, packing, and box-counting dimensions. The discussion is applied to a classic example such as the Peano curve.