Roger B. Nelsen

On the characterization of a class of binary operations on distribution functions

We characterize the class of binary operations \/o on distribution functions which are both induced pointwise, in the sense that the value of \/o(F, G) at g is a function of F(t) and G(t) (e.g. mixtures), and derivable from functions on random variables (e.g. convolution).

Best-possible bounds for the distribution of a sum ― a problem of Kolmogorov

SummaryRecently, in answer to a question of Kolmogorov, G.D. Makarov obtained best-possible bounds for the distribution function of the sumX+Y of two random variables,X andY, whose individual distribution functions,FX andFY, are fixed. We show that these bounds follow directly from an inequality which has been known for some time. The techniques we employ, which are based on copulas and their properties, yield an insightful proof of the fact that these bounds are best-possible, settle the question of equality, and are computationally manageable. Furthermore, they extend to binary operations other than addition and to higher dimensions.

Properties of a one-parameter family of bivariate distributions with specified marginals

Using a family of functions first described by Frank (1979), a one-parameter family of bivariate distributions is constructed. This family has arbitrary marginals and contains the Frechet bounds as well as the member corresponding to independent random variables, Three nonparametric measures of correlation (Spearmans rho, Ken-dalls tau, and the medial correlation coefficient) are evaluated, and a simple transformation to generate random samples from an ar-bitrary member of the family is presented.

Copulas and Association

A copula is a function of two variables which couples a bivariate distribution function to its marginal distribution functions. In doing so the copula captures certain nonparametric aspects of the relationship between the variates, from which it follows that measures of association and positive dependence concepts are properties of the copula. In this paper we survey results relating copulas to Spearman’s rho and Kendall’s tau for a variety of bivariate distributions, and we also show that certain positive dependence concepts (positive quadrant dependent, right tail increasing, left tail decreasing, and stochastically increasing) can be interpreted as simple geometric properties of the copula.

COPULAS CONSTRUCTED FROM DIAGONAL SECTIONS

If C is a copula, then its diagonal section is the function δ given by δ(t) = C(t, t). It follows that (i) δ(1) = 1; (ii) 0 ≤ δ(t 2) - δ(t 1) ≤ 2(t 2 - t 1) for all t 1, t 2 in [0,1] with t 1 ≤ t 2; and (iii) δ(t) ≤ t for all t in [0,1]. If δ is any function satisfying (i)-(iii), does there exist a copula C whose diagonal section is δ? We answer this question affirmatively by constructing copulas we call diagonal copulas: K(u, υ) = min(u, υ, (1/2)[δ(u) + δ(υ)]). We also present examples, investigate some dependence properties of random variables which have diagonal copulas, and answer an open question about tail dependence in bivariate copulas posed by H. Joe.

The Bertino Family of Copulas

Abstract In this paper we present some of the salient properties of the Bertino family of copulas. We describe the support set of a Bertino copula and show that every Bertino copula is singular. We characterize Bertino copulas in terms of the joint distribution of max (U,V) and min(U,V) when U and V are uniform [0,1] random variables whose copula is a Bertino copula. Finally, we find necessary and sufficient conditions for a Bertino copula to be extremal.

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.

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.

Concordance and Copulas: A Survey

Abstract In this paper we survey relationships between concordance of random variables and their copulas. We focus on the relationship between concordance and measures of association such as Kendall’s tau, Spearman’s rho and Gini’s coefficient. Extensions to the multivariate case are also discussed.

Some concepts of bivariate symmetry

Several concepts of bivariate symmetry are discussed, and relationships between and among these various concepts are explored. One of the concepts, radial symmetry, is shown to be nonparametric in the sense that it is a property of the copula of the random variables. Examples of distributions possessing the various types of symmetry are also presented.