Maurice J Frank

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.

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.

Generalization of independent response model for toxic mixtures

Interaction between toxic compounds has long been known to researchers. Attempts to model this interaction have been based on two basic paradigms--termed additivity and independence (1, 2). Previous models based on these assumptions focused on measuring the interaction between the compounds and then classifying the type of interaction as synergism, antagonism, additivity or independence (3, 4). The aim of this work is to present a generalization of the independent action hypothesis that is quantitatively capable of describing deviations regardless of the underlying single component dose response models. The mathematical framework of copulas is employed. This approach is then tested against data sets with both human health and ecological risk applications.

Convolutions for Dependent Random Variables

A two-parameter family σC,L of binary operations arises naturally as the distributional counterpart of operations on random variables. Here L is an arbitrary measurable function of two r.v.’s and C is their copula, or dependence function. Ordinary convolution is the particular member σProd,sum, i.e., when L is addition and C is the independence copula. In this paper, we investigate the algebraic and analytic structure of the σC,L, extending earlier results obtained for subfamilies.