José Antonio Rodríguez-Lallena

Bivariate copulas with cubic sections

In this paper we present a method for construction families of bivariate copulas with cubic cross sections. We study dependence properties, measures of association, and concepts of symmetry for these copulas. Examples of both symmetric and asymmetric copulas with cubic sections are presented which extend some well known families of bivariate copulas (such as the iterated Farlie-Gumbel-Morgenstern, Kimeldorf and Sampson, Lin, and Sarmanov families of copulas) and which provide second-order approximations to the Frank and Plackett families of copulas.

Some New Properties of Quasi-Copulas

Abstract The notion of a quasi-copula was introduced by Alsina et al. (1993) to characterize operations on distribution functions that can or cannot be derived from operations on random variables. Genest et al. (1999) characterize the quasi-copula concept in simpler operational terms. We present a new simple characterization and some properties of these functions, all of them concerning the mass distribution of a quasi-copula. We show that the features of this mass distribution can be quite different from that of a copula.

Best-Possible Bounds on Sets of Multivariate Distribution Functions

Abstract If H denotes the joint distribution function of n random variables X 1, X 2,…, X n whose margins are F 1, F 2,…, F n , respectively, then the fundamental best-possible bounds inequality for H is F 2(x 2),…, F n (x n )) for all x 1, x 2,…, x n in [−∞, ∞]. In this paper we employ n-copulas and n-quasi-copulas to find similar bounds on arbitrary sets of multivariate distribution functions with given margins. We discuss bounds for an n-quasi-copula Q when a value of Q at a single point is known. As an application, we investigate about bounds for a multivariate distribution function H with given univariate margins when the value of H is known at a single point whose coordinates are percentiles of the variables X 1, X 2,…, X n , respectively.

New constructions of diagonal patchwork copulas

We present a method for constructing symmetric copulas which generalizes the diagonal patchwork construction of copulas procedure. We also show how it is related to a new construction of a generalized Farlie-Gumbel-Morgenstern distribution and to the copula transforms.

Some advances in the study of the compatibility of three bivariate copulas

In this paper we study the problem of the compatibility of three bivariate copulas, i.e., we look for conditions which allow us to assure the existence of a three-copula whose two-dimensional margins are given. As a particular case, we seek conditions for two bivariate copulasC 1 andC 2 under whichC 2[C1 (x, y), z] is a three-copula. We specifically study the compatibility of the copulasM, W andII with other copulas both in general and in the particular case. We also study the compatibility of a two-copula with convex linear combinations of other two-copulas. Several examples illustrate the results obtained in each case, and some applications are given.

Some new characterizations and properties of quasi-copulas

In this paper we characterize the concept of quasi-copula in three different ways: as a special subclass of aggregation operators, in terms of its associated volume and-for the bivariate case-in terms of non-increasing tracks on [0,1]^2. We also provide sufficient conditions and new properties for quasi-copulas.

Quasi-copulas and signed measures

We study the relationship between multivariate quasi-copulas and measures that they may or may not induce on [0,1]^n. We first study the mass distribution of the pointwise best possible lower bound for the set of n-quasi-copulas for n>=3. As a consequence, we show that not every n-quasi-copula induces a signed measure on [0,1]^n.

Multivariate Archimedean Quasi-Copulas

Abstract In this paper we define and study basic properties of multivariate Archimedean quasi-copulas. In particular, we examine properties concerning generators, diagonal sections, permutation symmetry, level sets and order.

Bivariate quasi-copulas and doubly stochastic signed measures

We show that there exist bivariate proper quasi-copulas that do not induce a doubly stochastic signed measure on [0,1]^2. We construct these quasi-copulas from the so-called proper quasi-transformation square matrices.

Compatibility of Three Bivariate Quasi-Copulas: Applications to Copulas

In this paper we discuss the existence of trivariate copulas or quasi-copulas with prescribed three bivariate margins The case of quasi-copulas is completely solved. With respect to copulas, we provide a partial response to an interesting question posed in [8]. Several examples illustrate our results.