Manuel Úbeda-Flores

BOUNDS ON BIVARIATE DISTRIBUTION FUNCTIONS WITH GIVEN MARGINS AND MEASURES OF ASSOCIATION

We find pointwise best-possible bounds on the bivariate distribution function of continuous random variables with given margins and a given value of the population version of a nonparametric measure of association such as Kendalls tau or Spearmans rho.

Multivariate versions of Blomqvist’s beta and Spearman’s footrule

In this paper we define multivariate versions of the medial correlation coefficient and the rank correlation coefficient Spearman’s footrule in terms of copulas. We also present corresponding results for the sample statistic and provide a comparison of lower bounds among different measures of multivariate association.

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.

OPPOSITE DIAGONAL SECTIONS OF QUASI-COPULAS AND COPULAS

In this paper, we study opposite diagonal sections of quasi-copulas and copulas. The best-possible upper bound for the set of copulas with a given opposite diagonal section being known, we focus on the best-possible lower bound, which in general is a quasi-copula. Moreover, it exhibits an interesting type of bivariate symmetry called opposite symmetry.

A Comparison of Bounds on Sets of Joint Distribution Functions Derived from Various Measures of Association

Abstract We find pointwise best-possible bounds on the bivariate distribution function of continuous random variables with given margins and a given value of the medial correlation coefficient, and compare those bounds to those obtained from a given value of Kendall’s tau and Spearman’s rho.

Constructing copulas with given diagonal and opposite diagonal sections

A method for constructing copulas with given diagonal and opposite diagonal sections is presented. It makes use of a recently developed method for constructing cross-copulas with given horizontal and vertical sections. Conditions guaranteeing the existence of a cross-copula with the given diagonal and opposite diagonal sections are derived. It is shown how the new method facilitates the construction of families of copulas that simultaneously model tail dependences of upper-upper, upper-lower, lower-lower and lower-upper type.

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.

A copula-based family of fuzzy implication operators

A new family of fuzzy implication operators is introduced. The proposed class is based on the conditional version of a copula function. Properties of these operators are studied and several examples illustrate our results.