Berthold Schweizer

Associative functions : triangular norms and copulas

The functional equation of associativity is the topic of Abels first contribution to Crelles Journal. Seventy years later, it was featured as the second part of Hilberts Fifth Problem, and it was solved under successively weaker hypotheses by Brouwer (1909), Cartan (1930) and Aczel (1949). In 1958, B Schweizer and A Sklar showed that the “triangular norms” introduced by Menger in his definition of a probabilistic metric space should be associative; and in their book Probabilistic Metric Spaces, they presented the basic properties of such triangular norms and the closely related copulas. Since then, the study of these two classes of functions has been evolving at an ever-increasing pace and the results have been applied in fields such as statistics, information theory, fuzzy set theory, multi-valued and quantum logic, hydrology, and economics, in particular, risk analysis.This book presents the foundations of the subject of associative functions on real intervals. It brings together results that have been widely scattered in the literature and adds much new material. In the process, virtually all the standard techniques for solving functional equations in one and several variables come into play. Thus, the book can serve as an advanced undergraduate or graduate text on functional equations.

Thirty Years of Copulas

In 1959, in response to a query of M. Frechet, A. Sklar introduced copulas. These are functions that link multivariate distributions to their one-dimensional margins. Thus, if H is an n-dimensional cumulative distribution function with one-dimensional margins F1,…,Fn, then there exists an n-dimensional copula C (which is unique when F1,…,Fn are continuous) such that H(x1,…,xn) = C (F1(x1),…,Fn (xn)). During the years 1959 — 1974, most results concerning copulas were obtained in the course of the development of the theory of probabilistic metric spaces, principally in connection with the study of families of binary operations on the space of probability distribution functions. Then it was discovered that two-dimensional copulas could be used to define nonparametric measures of dependence for pairs of random variables. In the ensuing years the copula concept was rediscovered on several occasions and these functions began to play an ever-more-important role in mathematical statistics, particularly in matters involving questions of dependence, fixed marginals and functions of random variables that are invariant under monotone transformations. Today, in view of the fact that they are the higher dimensional analogues of uniform distributions on the unit interval, and as the result of the efforts of a diverse group of scholars, the significance, ubiquity and utility of copulas is being recognized. This paper is devoted to an historical overview and rather personal account of these developments and to a description of some recent results.

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.

Inequalities Among Operations on Probability Distribution Functions

Inequalities are established among certain binary operations on a space of probability distribution functions. These operations arise naturally in the theory of probabilistic metric spaces, in the generalized theory of information, and in other contexts of the theory of probability. It is further shown that, in most cases, equality holds in the inequalities if and only if at least one of the arguments is a unit step function, i.e., that the associated functional equations have in general only essentially trivial solutions.

The genesis of the notion of distributional chaos

The purpose of this paper is to indicate how the theory of distributional chaos was motivated by certain constructs from the theory of probabilistic metric spaces, to introduce the notion of distributional chaos and to illustrate some of its features with a simple example.

Bounds for distribution functions of sums of squares and radial errors

Bounds are found for the distribution function of the sum of squares X2+y2 where X and Y are arbitrary continuous random variables. The techniques employed, which utilize copulas and their properties, show that the bounds are pointwise best-possible when X and Y are symmetric about 0 and yield expressions which can be evaluated explicitly when X and Y have a common distribution function

On The τT-Product of Symmetric and Subsymmetric Distribution Functions

Let F be a non-defective distribution function (d.f.) and let \(\bar F\) be the d.f. defined by \(\bar F\,(x)\, = \,1\,\, - \,\,\ell ^ + F\,( - x)\) Then F is symmetric if \(F \le \bar F\); and we say that F is subsymmetric if \(F \le \bar F\). If T is a continuous t-norm and τT is the induced triangle function, then for any non-defective d.f.’s F and G we have \(\tau _T (\bar F,\bar G) \le \overline {\tau _T (F,G)}\). It follows that if F and G are subsymmetric then \(\tau \,_T \,(F,\,G)\,\) is subsymmetric. However, \(\tau \,_T \,(F,\,G)\,\) is symmetric for all symmetric F and G only if T = Min.