Copulas and quasi-copulas: an introduction to their properties and applications. (English) Zbl 1079.60021

Nelsen, Roger B.

Klement, Erich Peter (ed.) et al., Logical, algebraic, analytic and probabilistic aspects of triangular norms. Selected papers from the 24th Linz seminar on fuzzy set theory, Linz, Austria, February 4–8, 2003. Amsterdam: Elsevier (ISBN 0-444-51814-2/hbk). 391-413 (2005).

The aim of this paper is to survey the most important properties and applications of copulas and quasi-copulas. After the brief, introductory first Section, Sections 2–5 of the paper present the basic properties of copulas, and several families of copulas that are useful in statistical modeling. The class of Archimedean copulas, which are also triangular norms, is of particular interest. The author illustrates the great variety of such copulas, and reviews some of the most effective of their properties. Sections 6 and 7 explore the relationships among dependence concepts such as concordance, quadrant dependence, and likelihood ratio dependence, on one hand, and measures of association such as the population versions of Spearman’s rho, Kendall’s tau, and Gini’s gamma, on the other hand. Sections 8 and 9 discuss quasi-copulas and their relationship with copulas. The author considers some of the most interesting properties and applications of quasi-copulas, including recent results on the class of multivariate Archimedean quasi-copulas. Extensions to higher dimensions and a few open problems are outlined in the final Section 10.
For the entire collection see [Zbl 1063.03003].

Reviewer: Neculai Curteanu (Iaşi)

MSC:

60E05	Probability distributions: general theory
62H20	Measures of association (correlation, canonical correlation, etc.)

Keywords:

properties of copulas and quasi-copulas; Archimedean copulas; triangular norms; concordance; quadrant dependence; likelihood ratio dependence; multivariate Archimedean quasi-copulas; multivariate copulas and quasi-copulas