An introduction to copulas and maxmin copulas

If the distribution of a random vector is known, then the marginal distributions of its components are uniquely determined. On the other hand, univariate distributions can be joined into the multivariate one in numerous ways and copulas are one of the main tools in modelling the dependence of random variables. They join univariate distributions into the multivariate ones on the level of distribution functions. In the first part of the talk, we present the basics of the theory of copulas.

In the second part, we consider the so-called maxmin copulas that arise from the following binary shock model with one recovery option. Consider a system with components A and B which are subject to three different types of shocks. The first one is fatal for Component A only, the second one for Component B only, and the third type of shock affects both components simultaneously. Additionally, assume that Component A has a recovery option, or that we have an additional copy of Component A. The independent times of occurrences of three types of shocks are denoted respectively by X, Y and Z. Let U, respectively V, denote the lifetime of Component A, respectively Component B. The lifetime of Component B is expressed as V = min{Y,Z}, while the lifetime of Component A becomes U=max{X,Z}, since it is eliminated only by both types of shocks. The distribution of (U,V) is modelled by maxmin copula.