Copulas and dependence mea- surement

Copulas are a general tool for assessing the dependence structure of random variables. Important properties as well as a number of examples are discussed, including Archimedean copulas and the Marshall-Olkin copula. As measures of the dependence we consider linear correlation, rank correlation, the coefficients of tail dependence and association. Copulas are a tool for modeling and capturing the dependence of two or more random variables (rvs). In the work of [22] the term copula was used the first time; it is derived from the Latin word copulare, to connect or to join. Similar in spirit, Hoeffding studied distributions under “arbitrary changes of scale” already in the 1940s, see [7]. The main purpose of a copula is to disentangle the dependence structure of a random vector from its marginals. A d-dimensional copula is defined as a function C : [0, 1]d → [0, 1] which is a cumulative distribution function (cdf) with uniform1 marginals. On one side, this leads to the following properties: 1. C(u1, . . . , ud) is increasing in each component ui, i ∈ {1, . . . , d}; 2. C(1, . . . , 1, ui, 1, . . . , 1) = ui for all 1 ≤ i ≤ d; 3. For ai ≤ bi, 1 ≤ i ≤ d, C satisfies the rectangle 1Although standard, it is not necessary to consider uniform marginals (see copulas). inequality