A measure of mutual complete dependence

--

A scalar product for copulas

--

Gluing copulas

We present a new way of constructing bivariate copulas, by recalling and gluing two (or more) copulas. Examples illustrate how this construction can be applied to build complicated copulas from simple ones. --

Regression dependence

This paper presents a framework for comparing bivariate distributions according to their degree of regression dependence. We introduce the general concept of a regression dependence order (RDO) and provide two examples of RDOs. In addition, we define a new nonparametric measure of regression dependence and study its properties. Beside being monotone in the new RDOs, the measure takes on its extreme values precisely at independence and almost sure functional dependence, respectively. Finally, a consistent nonparametric estimator of the new measure is constructed

Regression dependence

This paper presents a framework for comparing bivariate distributions according to their degree of regression dependence. We introduce the general concept of a regression dependence order (RDO) and provide two examples of RDOs. In addition, we define a new nonparametric measure of regression dependence and study its properties. Beside being monotone in the new RDOs, the measure takes on its extreme values precisely at independence and almost sure functional dependence, respectively. Finally, a consistent nonparametric estimator of the new measure is constructed

A scalar product for copulas

We introduce a scalar product for n-dimensional copulas, based on the Sobolev scalar product for W 1,2 -functions. The corresponding norm has quite remarkable properties and provides a new geometric framework for copulas. We show that, in the bivariate case, it measures invertibility properties with respect to the ∗-product for copulas defined by Darsow et al. The unique copula of minimal norm is the null element for the ∗-multiplication, whereas the copulas of maximal norm are precisely the invertible elements

Almost opposite regression dependence in bivariate distributions

Let X,Y be two continuous random variables. Investigating the regression dependence of Y on X, respectively, of X on Y, we show that the two of them can have almost opposite behavior. Indeed, given any e > 0, we construct a bivariate random vector (X,Y) such that the respective regression dependence measures r2|1(X,Y), r1|2(X,Y) ∈ [0,1] introduced in Dette et al. (2013) satisfy r2|1(X,Y) = 1 as well as r1|2(X,Y) <e

A measure of mutual complete dependence

Two random variables X and Y are mutually completely dependent (m.c.d.) if there is a measurable bijection f with P(Y = f(X)) = 1. For continuous X and Y , a natural approach to constructing a measure of dependence is via the distance between the copula of X and Y and the independence copula. We show that this approach depends crucially on the choice of the distance function. For example, the Lp-distances, suggested by Schweizer and Wolff, cannot generate a measure of (mutual complete) dependence, since every copula is the uniform limit of copulas linking m.c.d. variables. Instead, we propose to use a modified Sobolev norm, with respect to which, mutual complete dependence cannot approx- imate any other kind of dependence. This Sobolev norm yields the first nonparametric measure of dependence capturing precisely the two extremes of dependence, i.e., it equals 0 if and only if X and Y are independent, and 1 if and only if X and Y are m.c.d. AMS 2000 subject classifcations: Primary 62E10; secondary 62H2

Symmetry of functions and exchangeability of random variables

We present a new approach for measuring the degree of exchangeability of two continuous, identically distributed random variables or, equivalently, the degree of symmetry of their corresponding copula. While the opposite of exchangeability does not exist in probability theory, the contrary of symmetry is quite obvious from an analytical point of view. Therefore, leaving the framework of probability theory, we introduce a natural measure of symmetry for bivariate functions in an arbitrary normed function space. Restricted to the set of copulas this yields a general concept for measures of (non-)exchangeability of random variables. The fact that copulas are never antisymmetric leads to the notion of maximal degree of antisymmetry of copulas. We illustrate our approach by various norms on function spaces, most notably the Sobolev norm for copulas

Gluing copulas

We present a new way of constructing bivariate copulas, by rescaling and gluing two (or more) copulas. Examples illustrate how this construction can be applied to build complicated copulas from simple ones